Chapter III Topological Spaces

1. Introduction

In Chapter I we looked at properties of sets, and in Chapter II we added some additional structure to a set
a distance function .
to create a pseudometric space. We then saw some of the most basic fundamental definitions and properties of pseudometric spaces. There is much more, and some of the most useful and interesting properties of pseudometric spaces will be discussed in Chapter IV. But in Chapter III we look at an important generalization.

We observed, early in Chapter II, that the idea of continuity (in calculus) depends on talking about “nearness,” so we used a distance function . to make the idea of “nearness” precise. In that way, we were able to extend the definition of continuity from ‘8 to pseudometric spaces. The distance function . also led us to the idea of open sets in a pseudometric space. From there we developed properties of closed sets, closures, interiors, frontiers, dense sets, continuity, and sequential convergence.

An important observation in Chapter II was that open (or closed) sets are really all that we require to talk about many of these ideas. In other words, we can often do what's necessary using the open sets without knowing which specific . generated the open sets: the topology g. is what really matters. For example, intE is defined in Ð\ß.Ñ in terms of the open sets ßE so int doesn't change if . is replaced with a different but equivalent metric .w
one that generates the same open sets. Similarly, changing . to an equivalent metric . w doesn't affect closures, continuity, or convergent sequences. In summary: for many purposes . is logically unnecessary once . has done the job of creating the topology g. (although having . might still be convenient).

This suggests a way to generalize our work. For a particular set \ we can simply assign a topology
that is, a collection of “open” sets given without any mention of a pseudometric that might have generated them. Of course when we do this, we want the “open sets” to “behave the way open sets should behave” as described in Theorem II.2.11. This leads us to the definition of a topological space.

2. Topological Spaces

Definition 2.1 A topology g on a set \ is a collection of subsets of \ such that

i) gß \ − g ii) if S−g for each α −Eß then S− g α +−E + iii) if SßÞÞÞßS"8 −g ß then S∩ÞÞÞ∩S " 8 − g Þ

A set S©\ is called open if S−g . The pair Ð\ßÑ g is called a topological space .

Sometimes we will just refer to “a topological space \ .” In that case, it is assumed that there is some topology g on \ but, for short, we are just not bothering to write down the “g ”)

103 We emphasize that in a topological space there is no distance function .: therefore phrases like “distance between two points” and “% -ball” make no sense in Ð\ß g ÑÞ There is no preconceived idea about what “open” means: to say “S is open” means nothing more or less than “S − g .”

In a topological space Ð\ßg Ñ , we can go on to define closed sets and isolated points just as we did in pseudometric spaces.

Definition 2.2 A subset J in Ð\ßg Ñ iscalled closed if X
J is open, that is, if \
J − g .

Definition 2.3 A point +−Ð\ßÑg is called isolated if Ö+ } is open, that is, if Ö+×−Þg

The proof of the following theorem is the same as it was for pseudometric spaces; we just take complements and apply properties of open sets.

Theorem 2.4 In any topological space Ð\ßg Ñ

i) g and \ are closed ii) if J is closed for each +−Eß then J is closed α α−E α iii) if JßÞÞÞßJ are closed, then 8 J is closed. " 83œ3 3 More informally, ii) and iii) state that intersections and finite unions of closed sets are closed.

Proof Read the proof for Theorem II.4.2. ñ

For a particular topological space Ð\ßg Ñ , it might or might not be possible to find a pseudometric .\ on that “creates” this topology
that is, one for which g. œÞ g

Definition 2.5 AtopologicalspaceÐ\ßg Ñ iscalledpseudometrizable ifthereexistsa pseudometric .\ on for which g. œÞ. g If is a metric, then Ð\ßÑ g is called metrizable.

Examples 2.6

1) Suppose \ is a set and g œÖgß\×Þ g is called the trivial topology on \ and it is the smallest possible topology on \ . Ð\ßg Ñ is called a trivial topological space. The only open (or closed) sets are g\Þ and If we put the trivial pseudometric .\ on , then g. œ g Þ So a trivial topological space turns out to be pseudometrizable.

At the opposite extreme, suppose gœ c Ð\Ñ . Then g is called the discrete topology on \ and it is the largest possible topology on \Þ Ð\ßg Ñ is called a discrete topological space. Every subset is open (and also closed). Every point of \ is isolated. If we put the discrete unit metric . (or any equivalent metric) on \œÞ , then g. g So a discrete topological space is metrizable.

104 2) Suppose \œÖ!ß"× and let g œÖgßÖ"×ß\× . Ð\ß g Ñ is a topological space called Sierpinski space . In this case it is not possible to find a pseudometric . on \ for which g. œ g , so Sierpinski space is not pseudometrizable. To see this, consider any pseudometric . on \ .

If .Ð!ß"Ñœ! , then . is the trivial pseudometric on \ and Ögß\לg. Á g .

If .Ð!ß"Ñœ$ ! , then the open ball FÐ!ÑœÖ!×−$ g. ß so g . Á g Þ ( In this caseg. is actually the discrete topology: . is just a rescaling of the discrete unit metric. )

Another possible topology on \œÖ!ß" } is gw œÖgßÖ!×ß\× , although Ð\ß g Ñ and Ð\ßg w Ñ seem very much alike: both are two-point spaces, each with exactly one isolated point. One space can be obtained from the other simply renaming “! ” and “" ” as “ " ”and “ ! ” respectively. Such “topologically identical” spaces are called “homeomorphic.” (We will give a precise definition later in this chapter.)

3) For a set \ , let g œÖS©\ÀSœg or \
S is finite ×Þ g is a topology on \À

i) Clearly, g−g and \− g .

ii)Suppose S−g for each α −EÞ If Sœgß then S− g . α α−Eα α −E α Otherwise there is at least one Sα! Ág . Then \
S α ! is finite, so \
Sœ Ð\
SÑ©\
S. Therefore \
S is α−Eα α −E αα! α −E α also finite, so S − g . α−E α iii) If SßÞÞÞßS−g and some Sœg , then 8 Sœg so 8 S− g . "8 33œ" 3 3œ" 3 Otherwise each S3 is nonempty, so each \
S3 is finite Þ Then Ð\
SÑ∪ÞÞÞ∪Ð\
Sќ\
8 S is finite, so 8 S− g . " 83œ" 3 3œ" 3 In Ð\ßÑJg , a set is closed iff JœgJ or is finite. Because the open sets are g and the co mplements of finite sets, g is called the cofinite topology on \ .

If \ is a finite set, then the cofinite topology is the same as the discrete topology on \ . (Why? ) If \ is infinite, then no point in Ð\ßÑg is isolated.

Suppose \ is an infinite set with the cofinite topologyg Þ If Y and Z are nonempty open sets, then \
Y and \
Z must be finite so Ð\
YÑ∪Ð\
ZÑ œ\
ÐY∩ZÑ is finite. Since \ is infinite, this means that Y∩ZÁgÐ in fact, Y ∩ Z must be infinite Ñ. Therefore every pair of nonempty opensets in Ð\ßg Ñ has nonempty intersection! This shows us that an infinite cofinite space Ð\ßg Ñ is not pseudometrizable:

i) if . is the trivial pseudometric on \ , then certainly g. Á g , and

ii) if . is not the trivial pseudometric on \ , then there exist points

+ß,−\ for which .Ð+ß,Ñœ$ ! . In that case, F$Î# Ð+Ñ and FÐ,Ñ$Î# would be disjoint nonempty open sets in g.ß so g . Á g Þ

105 4) On ‘g , let œÖÐ+ß∞Ñ : +− ‘‘ ×∪Ög , × . It is easy to verify that g is a topology on ‘ , called the right-ray topology . Is Ð ‘ ß g Ñ metrizable or pseudometrizable?

If \œg , then g œÖg× is the only possible topology on \ , and g œÖgßÖ+×× is the only possibletopology on a singleton set \œÖ+× . But for l\l" , there are many possible topologies on \ . For example, there are four possible topologies on the set \ œ Ö+ß,× . These are the trivial topology, the discrete topology, ÖgßÖ+×ß\×ß and ÖgßÖ,×ß\× although, as mentioned earlier, the last two can be considered as “topologically identical.”

If g is a topology on \ , then g is a collection of subsets of \ß©Ð\Ñ so gc . This means that g− cc Ð Ð\ÑÑ, so | cc Ð Ð\ÑÑœ# |Ð#l\l Ñ is an upper bound for the number of possible topologies on \ . For example, there are no more than #¸$Þ%‚"!#( $) topologies on a set \ with 7 elements. But this upper bound is actually very crude, as the following table (given without proof) indicates: 8 œ l\l Actual number of distinct topologies on \

0 1 1 1 2 4 3 29 4 355 5 6942 6 209527 7 9535241 (many less than "! $) )

Counting topologies on finite sets is really a question about combinatorics and we will not pursue this topic.

Each concept we defined for pseudometric spaces can be carried over directly to topological spaces if the concept was defined in topological terms
that is ß in terms of open (or closed) sets. This applies, for example, to the definitions of interior, closure, and frontier in pseudometric spaces, so these definitions can also be carried over verbatim to a topological space Ð\ßg ÑÞ

Definition 2.7 Suppose E©Ð\ßg Ñ . We define

int the interior of in is open and } \E œ E \ œ ÖSÀS S©E× cl the closure of in is closed and \E œ E \ œÖJÀJ J ª E× Fr\E œ the frontier (or boundary) of E in \ œ cl\E∩ cl \ Ð\
EÑ

As before, we will drop the subscript “\ ” when the context makes it clear.

The properties for the operators cl, int, and Fr (except those that mention a pseudometric . or an % -ball) remain true. The proofs in the preceding chapter were deliberately phrased in topological terms so they would carry over to the more general setting of topological spaces.

106 Theorem 2.8 Suppose E © Ð\ß.Ñ . Then

1) a) int E is the largest open subset of E (that is, if S is open and O ©E , then S ©int E ). b) E is open iff EœEÐ int Note: int E©E is true for every set E so we could say: E is open iff E © int EÞÑ c) B−E int iff there is an open set S such that B−©E S

2) a) cl E is the smallest closed set containing EJJE (that is, if is closed and ª , then J ª cl E ). b) E is closed iff EœEÐE©E cl Note: cl is true for every set E , so we could say: E is closed iff cl E©EÞÑ c) B−E cl iff for every open set S containing B∩EÁg , S

3) a) FrE is closed and Fr Eœ Fr(X
E ). b)B−E Fr iff for every open set S containing B∩EÁg , S and S∩Ð\
EÑ Á g cÑE is clopen iff Fr Eœg .

See the proof of Theorem II.4.5

At this point, we add a few additional facts about these operators. Some of the proofs are left as exercises.

Theorem 2.9 Suppose EF , are subsets of a topological space Ð\ßÑÞg Then

1) clÐE∪Ñœ F cl ÐE∪ Ñ cl ÐF Ñ 2) clEœE∪ Fr E 3) intEœE
Fr Eœ\
cl Ð\
EÑ 4) FrEœ cl E
int E 5) \œ int E∪ Fr E∪ int ÐX
E Ñ , and these 3 sets are disjoint.

Proof 1) E © E ∪ F so, from the definition of closure, we have cl E©cl ÐE∪FÑ . Similarly, clF© cl ÐE∪FÑ Therefore cl E∪ cl F© cl ÐE∪FÑÞ On the other hand, clE∪F cl is the union of two closed sets, so cl E∪F cl is closed and clE∪ cl FªE∪F , so cl E∪ cl Fª cl ÐE∪FÑ . ( As an exercise, try proving 1) instead using the characterization of closures given above in Theorem 2.8.2c. ) Is 1) true if “∪ ” is replaced by “ ∩ ”?

2) Suppose B−E cl but BÂEÞ If S is any open set containing B , then S∩EÁg (because B−E cl ) and S∩Ð\
EÑÁgÐ because the intersection contains B ). Therefore B−Fr E , so B−A ∪ Fr A Þ Conversely, suppose B−E∪ Fr E . If B−E , then B− cl EÞ And if BÂE , then B−Fr Eœ cl E∩Ð\
EÑ cl , so B−EÞ cl Therefore clA œ A ∪ Fr A Þ

The proofs of 3)
5) are left as exercises. ñ

107 Theorem 2.9 shows us that complements, closures, interiors and frontiers are interrelated and therefore some of these operators are redundant. That is, if we wanted to very “economical,” we could discard some of them. For example, we could avoid using “Fr” and “int” and just use “cl” and complement because FrEœ cl E∩ cl Ð\
EÑ and int EœE
Fr E œE
Ðcl E∩ cl Ð\
EÑÑ . Of course, the most economical way of doing things is not necessarily the most convenient. ( Could we get by only using complements and “Fr”
that is, can we define “int” and “”cl” in terms of “Fr” and complements? Or could we use just “int” and complements? )

Here is a famous related problem from the early days of topology: for E©Ð\ßg Ñ , is there an upper bound for the number of different subsets of \ which might created from E using only complements and closures , repeated in any order? ( As we just observed, using the interior and frontier operators would not help to create any additional sets. ) For example, one might start with E and then consider such sets as cl E\
EßÐ\
Ð\
EÑÑß , cl cl cl and so on. An old theorem of Kuratowski (1922) says that for any set E in any space Ð\ßÑg , the upper bound is 14. Moreover, this upper bound is “sharp”
there is a set E © ‘ from which 14 different sets can actually be obtained! Can you find such a set?

Definition 2.10 Suppose Ð\ßÑg is a topological space and H©\H . is called dense in \ if clH œ \Þ The space Ð\ßg Ñ is called separable if there exists a countable dense set H in \ .

Example 2.11 Let g be the cofinite topology on ‘ . Let „ œ Ö#ß%ß'ßÞÞÞ×Þ

int„ œ g , because each nonempty open set S has finite complement and therefore S©IÞ// In fact, for any F©À‘ if ‘
F is infinite, then int Fœg / cl„‘œ because the only closed set containing „‘ is . Therefore Ð ‘g ß Ñ is separable. In fact, any infinite set F is dense.

Fr„œ cl ÐÑ∩ „ cl Ð ‘„‘‘‘
Ñœ ∩ œ

Б ß g Ñ is not pseudometrizable (why? )

Example 2.12 Let g be the right-ray topology on ‘‘g . In (ß Ñ ,

int ™ œ g

cl™œ ‘ , so ÐßÑ ‘g is separable

Fr ™œ ‘

Any two nonempty open sets intersect, so Б ß g Ñ is not metrizable. Is it pseudometrizable?

108 3. Subspaces

There is a natural way to create a new topological space from a subset of a given topological space. The new space is called a subspace (not merely a subset ) of the original space.

Definition 3.1 Suppose Ð\ßÑg is a topological space and that E©\ . The subspace topology on E is defined as gE œÖE∩SÀS− gg } and ÐEßE Ñ is called a subspace of Ð\ßÑÞ g We sometimes call S∩E the “restriction of SE to ” or the “trace of SE on ”

You should check the conditions in Definition 2.1 are satisfied: in other words, that gE really is a topology on E. When we say that E is a subspace of Ð\ßÑg , we mean that E is a subset of \ with the subspace topology gE Þ To indicate that E is a subspace, we sometimes write E©Ð\ßg Ñrather than E©\ Þ

Example 3.2 Consider ‘‘© , where has its usual topology. For each 8 − , the interval Ð8
"ß81"Ñ is open in ‘ . Therefore Ð8
"ß81"Ñ∩ œÖ8× is open in the subspace , so every point 8 is isolated in the subspace. The subspace topology is the discrete topology. Notice: the subspace topology on is the same as what we get if we use the usual metric on to generate open sets in . Similarly, it is easy to check that in ‘ # the subspace topology on the B - axis is the same as the usual metric topology on ‘ .

More generally: suppose E©\ , where Ð\ß.Ñ is a pseudometric space. Then we can think of two ways to make E into a topological space.

i) . gives a topology g. on \ . Take the open sets in g . and intersect them with E . This gives us the subspace topology on E , which we call Ðg. Ñ E and ÐEßÐÑÑg. E is a subspace of Ð\ßÑÞ g

2) Or, we could view E as a pseudometric space by using . to measure distances in E.À\‚\Ä . To be very precise, ‘ , so we make a “new” pseudometric .w defined by .œ.lE‚E w . Then ÐEß.Ñ w is a pseudometric space and we can w use . to generate open sets in E : the topology g.w Þ Usually, we would be less compulsive about notation and continue to use the name “d” also for the pseudometric on E . But for a moment it will be helpful to distinguish carefully between . and .w œ.lE‚E .

Fortunately, it turns out that 1) and 2) produce the same open sets in EÀ the open sets in ÐEß.Ñw are just the open sets from Ð\ß.Ñ restricted to EÞ That's just what Theorem 3.3 says, in “fancier” notation.

Theorem 3.3 Suppose E©\ß where Ð\ß.Ñ is a pseudometric space. Then ÐÑœg. E g . w .

Proof If Y−ÐÑg.E , then YœS∩E where S− g. . Let +−Y . There is an % ! such that . w ..w FÐ+Ñ©SÞ% Since . œ.lE‚E , we get that F% Ð+ÑœFÐ+Ñ∩E©S∩EœY % , so Y− g. w . Conversely, suppose Y−Þg. w For each +−Y there is an % + ! such that . w .w . FÐ+Ñ©Yß and Yœ FÐ+ÑÞ Let Sœ FÐ+Ñß an open set in g. Þ Since %+ +−Y%+ +−Y % + F..w Ð+ÑœF Ð+Ñ∩E, we get S∩EœÐ F. Ð+ÑÑ∩EœÐ F . Ð+Ñ∩EÑ %%+ + +−S%+ +−S % + .w œ FÐ+ÑœYÞ Therefore Y−ÐÑÞñg. E +−Y %+

109 Exercise Verify that in any topological space Ð\ßg Ñ

i) If Y is open in Ð\ßÑEg and is an open set in the subspace YßE then is open in \Þ (“An open subset of an open set is open.” ) ii) If J is closed in Ð\ßÑEg and is a closed set in the subspace JßJ then is closed in \ . (“A closed subset of a closed set is closed.” )

4. Neighborhoods

Definition 4.1 Let Ð\ßÑg be a topological space and suppose R©\ÞB−R If int , then we say that N is a neighborhood of B Þ The collection aB œÖR©\ À R is a neighborhood of B× is called the neighborhood system at BÞ

Note that 1) aB Ág , because every point B has at least one neighborhood
for example, \ −aB Þ 2) If R"#B and R−a , then B−R∩Rœ int "# int (why? ) int ÐR∩RÑ "# . Therefore R"∩ R # − a B . w w w 3) If R−aB and R©R, then B− int R© int R , so R− a B (that is, if Rw contains a neighborhood of BR , then w is also a neighborhood of B .)

Just as in pseudometric spaces, it is clear that a set S in Ð\ßg Ñ is open iff O is a neighborhood of each of its points . (See Theorem II.5.4 )

In a pseudometric space, we use the % -balls centered at B to measure “nearness” to B . For example, if “every % -ball in ‘ centered at B contains an irrational number,” this tells us that “there are irrational numbers arbitrarily near to B .” Of course, we could convey the same information in terms of neighborhoods by saying “every neighborhood of B in ‘ contains an irrational number.” Or, instead, we could say it in terms of “open sets”: “every open set in ‘ containing B contains an irrational number.” These are all equivalent ways to say “there are irrational numbers arbitrarily near to B .” This isn't that surprising sinceopen sets and neighborhoods in Ð\ß .Ñ were defined in terms of % -balls.

In a topological spaceÐ\ßg Ñ we don't have % -balls, but we still have open sets and neighborhoods. We now think of the neighborhoods in aB (or, if we prefer, the collection of open sets containing B ) as the tool we use to talk about “nearness to B .”

For example, suppose \ has the trivial topology g . For any B−\ß the only neighborhood of B is \ : therefore every C\ in is in every neighborhood of B À the neighborhoods of B are unable to “separate” B and C , and that's analogous to having .ÐBßCÑœ! (if we had a pseudometric). In that sense, all points in Ð\ßg Ñ are “very close together”: so close together, in fact, that they are “indistinguishable.” The neighborhoods of B tell us this.

At the opposite extreme, suppose \ has the discrete topology g and that B−\ÞB−[ If then (since [ is open ), [ is a neighborhood of B . The smallest neighborhood of BRœÖB× is . So every point B has a neighborhood R that excludes all other points CCÁB : for every , we could

110 say “C is not within the neighborhood RB of .” This is analogous to saying “ C is not within % of B” (if we had a pseudometric). Because no point C is “within RBß of ” we call B isolated. The neighborhoods of B tell us this.

Of course, if we prefer, we could use “Y is an open set containing B ” instead of “ R is a neighborhood of B ” to talk about nearness to B .

The complete neighborhood system aB of a point B often contains more neighborhoods than we actually need to talk about nearness to B . For example, the open balls FÐBÑ" in a pseudometric 8 space Ð\ß .Ñ are enough to let us talk about continuity at B . Therefore, we introduce the idea of a neighborhood base at B to choose a smaller collection of neighborhoods of B that is

i) good enough for all our purposes, and ii) from which all the other neighborhoods of B can be obtained if we want them.

Definition 4.2 A collection UB© a B is called a neighborhood base at B if for every neighborhood RB of , there is a neighborhood F−UB such that BF©R− . We refer to the sets in UB as basic neighborhoods of B.

According to the definition, each set in UB must be a neighborhood of B , but the collection U B may be much simpler than the whole neighborhood system RB Þ The crucial thing is that every neighborhood RB of must contain a basic neighborhood FB of .

Example 4.3 In Ð\ß .Ñ , possible ways to choose a neighborhood base at B include:

i) UB œ the collection of all balls FÐBÑ% , or ii) UB œ the collection of all balls FÐBÑ% , where % is a positive rational, or iii) UB œ the collection of all balls FÐBÑ−" for 8 , or 8 iv) UBœ a B ( The neighborhood system is always a base for itself, but it is not an “efficient”choice; the goal is to get a base UB that's much simpler than RB.)

Which UB to use is our choice: each of i)-iv) gives a neighborhood base at B . But ii) or iii) might be more convenient
because ii) and iii) are countable neighborhood bases UB . ( If \ œ ‘, for example, with the usual metric . , then the collections U B in i) and iv) are uncountable. ) Of the four, iii) is probably the simplest choice for UB .

Suppose we want to check whether some property that involves neighborhoods of B is true. Often all we need to do is to check whether the property holds for neighborhoods in the simpler collection UB . For example, in Ð\ß .Ñ À S is open iff O contains a neighborhood R of each B−SÞ But that is true iff S contains a set FÐBÑ" around each of its points B . 8

Similarly, B−E cl iff R∩AÁg for every R− aB iff F∩EÁg for every F− UB . For example, suppose we want to check, in ‘ , that " − cl . It is sufficient just to check that FÐ"Ñ∩Ág" for each 8− , because this implies that R∩Ág for every R−Þ a " 8

Therefore it's often desirable to make an “efficient” choice of neighborhood base UB at each point B − \ .

111 Definition 4.4 We say that a space Ð\ßg Ñ satisfies the first axiom of countability (or, more simply, that Ð\ßÑg is first countable ) if at each point B−\ , it is possible to choose a countable neighborhood base UBÞ

Example 4.5

1) The preceding Example 4.3 shows that every pseudometric space is first countable.

2) If g is the discrete topology on \ß then Ð\ßg Ñ is first countable. In fact, at each point B , we can choose a neighborhood base that consists of a single set: UB œ ÖÖB××Þ

3) Let g be the cofinite topology on an uncountable set \ . For any B−\ , there cannot be a countable neighborhood base UB at B .

We prove this by contradiction: suppose we had a countable neighborhood base at some point BÀ call it UB œÖFßÞÞÞßFßÞÞÞ×Þ " 8

For any CÁBßÖC× is closed ß so \
ÖC× is a neighborhood of B . Then, by the definition of neighborhood base, there is some 5 for which B− int F©F5 5 ©\
ÖC× . Therefore CÂ∞ int F . 8œ" 8 But B−∞ int F so ∞ int FœÖB× . 8œ"8 8œ" 8

Then \
ÖBל\
∞ int Fœ ∞ Ð\
int FÑ . Since \
int F is 8œ"8 8œ" 8 8 finite (why ?),this would mean that \
ÖB× is countable
which is impossible.

Since any pseudometric space is first countable, the example gives us another way to see that this space Ð\ßg Ñ is not pseudometrizable.

In Ð\ßg Ñ , the neighborhood system a B at each point B is completely determined by the topology g , but U B is not. As the preceding examples illustrate, there are usually many possible choices for UB . ( Can you describe all the spaces Ð\ßg Ñ for which U B is uniquely determined at each point B ?)

On the other hand, if we were given UB at each point B − \ we could

1) “reconstruct” the whole neighborhood system aB :

aBœÖR©\ÀbF− U B such that B−F©R× , and then we could

2) “reconstruct” the whole topology g :

g œÖSÀS is a neighborhood of each of its points × œÖSÀaB−SbF−UB B−F©S× , that is, S is open iff S contains a basic neighborhood of each of its points.

This illustrates one method of describing a topology: by telling the neighborhood basis UB at each point. Various effective methods to describe topologies are discussed in the next section.

112 5. Describing Topologies

How can a topological space be described? If \ œ Ö!ß"× , it is simple to give a topology by just writing g œ ÖgßÖ!×ß\×Þ However, describing all the sets in g explicitly is often not the easiest way to go.

In this section we look at three important
different but closely related
ways to define a topology on a set. All of the7 will be used throughout the course. A fourth method
by using a “closure operator”
is not used much nowadays. It is included just as an historical curiosity.

A. Basic neighborhoods at each point

Suppose that at each point B−Ð\ßg Ñ we have picked a neighborhood base U B . As mentioned above, the collections UB implicitly contain all the information about the topology: a set S is in g iff O contains a basic neighborhood of each of its points. This suggests that if we start with just a set \ , then we could define a topology on \ if we begin by saying what the UB 's should be. Of course, we can't just put “random” sets in UBÀ the sets in each U B must “act the way basic neighborhoods are supposed to act.” And how is that? The next theorem describes the crucial behavior of a collection of basic neighborhoods at B in any topological space.

Theorem 5.1 Suppose Ð\ßg Ñ is a topological space and that for each point B − \ß U B is a neighborhood base.

1) UBÁg and F− U B ÊB−F©\

2) if F"# and F−UB , then bF− $ U B such that B−F $© F " ∩F #

3) if F−UB , then bM such that B − M © F and, a C−b−I, BCU C such that C −© BI C

4) O−ÍaB−b−g OB U B such that B − BO © .

Proof 1) Since \ is a neighborhood of B , there is a F−UB such that B − F © \ . Therefore UBÁg−©. If F U B a B , then B is a neighborhood of BB , so − int F ©F©\ .

2) The intersection of the two neighborhoods FFB" and # of is a neighborhood of B . Therefore, by the definition of a neighborhood base, there is a set F$ − U B such that B − F $© F " ∩F # .

3) Let Mœ int F. Then B − IB © and because I is open, I is a neighborhood of each its points C . Since UC is a neighborhood base at C, there is a set FC −C−FU C such that C © M .

4) Ê : If O is open, then O is a neighborhood of each of its points B . Therefore for each B−S there must be a set F−UB such that B−F©SÞ É :The condition implies that O contains a neighborhood of each of its points. Therefore S is a neighborhood of each of its points, so S is open. ñ

113 Theorem 5.1 lists the crucial features of the behavior of a neighborhood base at B . The next theorem tells us that we can put a topology on a set \ by assigning a “properly behaved" collection of sets to become the basic neighborhoods at each point B .

Theorem 5.2 (The Neighborhood Base Theorem) Let \ be a set . Suppose that for each B−\ we give a collection UB of subsets of \ in such a way that conditions 1) - 3) of Theorem 5.1 are true.Define gœ {O©\ : aB−bF− O UB such that B −F© O } Þ Then g is a topology on \ and UB is now a neighborhood base at BÐ\ßÑÞ in g

Note: In Theorem 5.2, we do not ask that the UB 's satisfy condition 4) of Theorem 5.1
since \ is a set with no topology (yet), condition 4) would be meaningless. Rather, Condition 4) becomes the motivation for how to define a topology g using the FB 's.

Proof We need to prove three things: a) g is a topology b) each F − UB is now a neighborhood of B in Ð\ßg Ñ , and c) the collection UB is now a neighborhood base at B .

a) Clearly, g −g . If B − \ then, by condition 1), we can choose any F − U B and B−F©\Þ Therefore \ − g .

Suppose O−g for all α −EB− . If { O : α −E }, then B− O for some α α α ! α−E. By definition of gU ß there is a set F − such that B −B © O ©−E { O : α }, so ! B α! α {:O α } g . α − E −

To finish a), it is sufficient to show that if S"and S # −g , then S∩S" # − g . Suppose B − S" ∩ S # . By the definition of g , there are sets F " and F # − U B such that B©B−F""# S and −F©S# , so B −F∩F©S∩S "#" # . By condition 2), there is a set F−$U B such that B©F−F $ "#"# ∩F©∩OO . Therefore OO "# ∩ − g .

Therefore g is a topology on \
so now we have a topological space Ð\ßÑ
g and we must show that UB is a neighborhood base at B in Ð\ß g Ñ . Doing so involves the awkward-looking condition 3)
which we have not yet used.

b) If F−UB , then B − F (by condition 1) and (by condition 3), there is a set M © X such that B−M©F and a C − MF, b C − UC such that C − FMC © . The underlined phrase states that M satisfies the condition for M−g , so M is open. Since M is open and B −M©FF, is a neighborhood of B , that is, UB © a B .

c) To complete the proof, we have to check that UB is a neighborhood base at B . If R is a neighborhood of BB−R , then int . Since int R is open, int R must satisfy the criterion for membership in g , so there is a set F− UB such that B− F ©R©R int . Therefore U B forms a neighborhood base at B . ñ

114 Example 5.3 For each B−‘ , let U B œÖÒBß,ÑÀ,B× . We can easily check the conditions 1) - 3) from Theorem 5.2:

1) For each B−‘U , certainly B Ág and B−ÒBß,Ñ for each set ÒBß,Ñ− U B 2) If F" œÒBß,Ñ " and F # œÒBß,Ñ # are in U B , then (in this example) we can choose FœF∩FœÒBß,Ñ−$"# $BU , where ,œ $ min Ö,ß,×Þ "# 3) If FœÒBß,Ñ−UB , then (in this example) we can let MœFÞ If C−MœÒBß,Ñß pick - so C@-@, . Then FC œÒCß-Ñ− U C and C − ÒCß-Ñ © MÞ

According to Theorem 5.2, gœÖS©ÀaB−S ‘bÒBß,Ñ − U B such that B −ÒBß,Ñ©S× is a topology on ‘U and B is a neighborhood base at BÐßÑ in ‘g . The space ÐßÑ ‘g is called the Sorgenfrey line .

Notice that in this example each set ÒBß ,Ñ − g : that is, the sets in U B turn out to be openß not merely neighborhoods of BÞÐThis does not always happen. Ñ

It is easy to check that sets of form Ð
∞ßBÑ and Ò,ß∞Ñ are open, so Ð
∞ßBÑ∪Ò,ß∞Ñ œ
ÒBß,Ñ‘ is open. Therefore ÒBß,Ñ is also closed. So at each point B in the Sorgenfrey line, there is a neighborhood base UB consisting of clopen sets.

We can write Ð+ß-Ñœ∞ Ò+1" ß-Ñ , so Ð+ß-Ñ is open in the Sorgenfrey line. Because every 8œ" 8 usual open set in ‘ is a union of sets of the form Ð+ß -Ñ , we conclude that every usual open set in ‘ is also open in the Sorgenfrey line . The usual topology on ‘ is strictly smaller that the Sorgenfrey topology: g. § g . Á

is dense in the Sorgenfrey line: if B− ‘ , then every basic neighborhood ÒBß,ÑB of intersects , so B − cl Þ Therefore the Sorgenfrey line is separable. It is also clear that the Sorgenfrey " line is first countable: at each point B the collection ÖÒBßB1ÑÀ8−×8 is a countable neighborhood base.

Example 5.4 Similarly, we can define the Sorgenfrey plane by putting a new topology on ‘# . # At each point ÐBßCÑ−‘U , let ÐBßCÑ œÖÒBß,Ñ‚ÒCß-ÑÀ,Bß-C× . The families U ÐBßCÑ satisfy the conditions in the Neighborhood Base Theorem, so they give a topology g for which # UÐBßCÑ isa neighborhood base at ÐBßCÑÞ A set S©‘ is open iff: for each ÐBßCÑ−S , there are ,B and -C such that ÐBßCÑ−ÒBß,Ñ‚ÒCß-Ñ©SÞÐ Make a sketch! Ñ You should check that the sets ÒBß,Ñ ‚ ÒCß-Ñ − UÐBßCÑ are actually clopen in the Sorgenfrey plane. It is also easy to check the usual topology g. on the plane is strictly smaller that the Sorgenfrey topology. It is clear that # is dense, so Ð ‘ # ß g Ñ is separable. Is the Sorgenfrey plane first countable?

# # Example 5.5 At each point :−‘ , let GÐ:ÑœÖB−% ‘ À.ÐBß:ÑŸ % × and define # U: œÖGÐ:ÑÀ% :− ‘ × . It is easy to check that the conditions 1)-3) of Theorem 5.2 are satisfied. ÐFor GÐ:Ñ% in condition 3), let I œFÐ:ÑÞÑ% The topology generated by the U: 's is just the usual topologyà the sets in U: are basic neighborhoods of : in the usual topology
as they should be
but the sets in U: did not turn out to be open sets.

115 Example 5.6 Let >œÖÐBßCÑ− ‘ # ÀC !ל the “closed upper half-plane.”

For a point :œÐBßCÑ−> with C!ß let U: œÖFÐ:ÑÀ% % @lCl× For a point :œÐBß!Ñ−> ß let

U: œ ÖÖ:× ∪ E À E is a usual open disc in the upper half-plane, tangent to the B -axis at :×Þ

It is easy to check that the collections U: satisfy the conditions 1) -3) of Theorem 5.2 and therefore give a topology on > . In this topology, thesets in U : turn out to be open neighborhoods of : .

The space > , with this topology, is called the “Moore plane .” Notice that > is separable and first countable. The subspace topology on the B -axis is the discrete topology. ( Verify these statements! )

B. Base for the topology

Definition 5.7 A collection of open sets in Ð\ßg Ñ is called a base for the topology g if each S−g is a union of sets from U Þ More precisely, UUg is a base if © and for each S−ß g there exists a subfamily TU such that T . We also call U a base for the open sets and © S œ we refer to the open sets in U as basic open sets .

If U is a base, then it is easy to see that: S−g iff aB−O bF− U such that BF− © O . This means that if we were given U , we could use it to decide which sets are open and thus “reconstruct” g .

Of course, one example of a base is Uœ g : every topology g is a base for itself. But usually there are many ways to choose a base, and the idea is that a simpler base U would be easier to work with. For example, the set of all balls is a base for the topology g. in any pseudometric space Ð\ß .Ñà a different base would be the set containing only the balls with positive rational radii . (Can you describe those topological spaces for which g is the only base for g ?)

The following theorem tells us the crucial properties of a base U in Ð\ß g ÑÞ

Theorem 5.8 If Ð\ßg Ñ is a topological space with a base U for g , then

1) U \œ ÖFÀF− ×

2) if FF−"# and U and BFF−" ∩ # , then there is a set F− $ U such thatB−F$ ©F "∩ F # .

Proof 1) Certainly U and since is open the definition of base implies that is the ©\
\
\ union of a subfamily of U . Therefore U œ \Þ

2) If F"# and F−U , then F∩F "# is open so F∩F "# must be the union of some sets from U . Therefore, if BFF−"# ∩ , there must be a set F−$ U such that BFFFñ−$ © " ∩ # .

116 The next theorem tells us that if we are given a collection U of subsets of a set \ with properties 1) and 2), we can use it to define a topology.

Theorem 5.9 (The Base Theorem) Suppose \ is a set and that U is a collection of subsets of \ that satisfies conditions 1) and 2) in Theorem 5.8. Define g œ Ö O© \ À O is a union of sets from UלÖS©\ÀaB−SbF− U such that B−F©S× . Then g is a topology on \ and U is a base for g .

Proof First we show that g is a topology on \ . Since g is the union of the empty subfamily of U, we get that g− g , and condition 1) simply states that \− g .

If O g ( α ), then α is a union of sets each of which is itself a union of α− −E ÖSÀ−E× α S α sets from U . Then clearly α is a union of sets from Uαg , so . ÖSÀ−E×α ÖSÀ−E×−α

Suppose O" and O #−g and that B−S" ∩ O # . For each such B we can use 2) to pick a set F−BU such that B−F B © OO" ∩ # . Then OO " ∩ # is the union of all the B B 's chosen in this way, so O" ∩ O # − g .

Now we know that we have a topology, g , on \ . By definition of g it is clear that Ug © and that each set in g is a union of sets from UUgÞ Therefore is a base for . ñ

Example 5.10 The collections

# U œÖFÐÑÀ:−: % ‘%, !× w" # U œÖFÐÑÀ:−: 5 , 5− × and Uww œÖÐ+ß,Ñ‚Ð-ß.ÑÀ+ß,ß-ß.− ‘ , +@,ß-@.× each satisfy the conditions 1) - 2) in Theorem 5.9, so each collection is the base for a topology on ‘#Þ In fact, all three are bases for the same topology on ‘ #
that is, the usual topology Ð check this! Ñ Of the three, U w is the simplest choice
it is a countable base for the usual topology.

Example 5.11 Suppose Ð\ßÑÐ]ßÑgw and g ww are topological spaces. Let U œ the set of “open boxes” in \‚]œÖY‚ZÀY−gw and Z− g ww ×Þ ( Verify that U satisfies conditions 1) and 2) of The Base Theorem. ) The product topology on the set \ ‚ ] is the topology for which U is a base. We always assume that \ ‚ ] has the product topology unless something else is stated.

Therefore a set E©\‚] is open (in the product topology) iff À for all ÐBßCÑ−Eß there are open sets Y©\ and Z©] such that ÐBßCÑ−Y‚Z©EÞ Ð Note that E itself might not be a “box.” Ñ

Let 1" À\‚]Ä\ be the “projection” defined by 1 " ÐBßCÑœBY . If is any open set in \ ,
"
" then 1" ÒYÓœY‚]− U1 . Therefore " ÒYÓ is open. Similarly, for 1 # À\‚]Ä] defined
" by 1#ÐBßCÑœCÀ if Z is open in ] , then 1 # ÒZÓœ\‚Z is open in \‚]Þ ( As we see in Section 8, this means that the projection maps are continuous. It is not hard to show that

117 the projection maps 1" and 1 # are also open maps: that is, the image of open sets in the product is open ).

If H"# is dense in \H and is dense in ] , we claim that H‚H" # is dense in \‚] . If ÐBß CÑ − E, where E is open, thenthere are nonempty open sets Y©\ and Z©] for which ÐBßCÑ−Y‚Z©EÞ Since B− cl H" , we know that Y∩HÁg " ; and similarly Z∩HÁgÞ # Therefore ÐY‚ZÑ∩ÐH"# ‚HÑœÐY∩HÑ‚ÐZ∩HÑÁg " #, so E∩ÐH "# ‚HÑÁgÞ Therefore ÐBßCÑ− cl ÐH‚HÑÞ"# So H‚H "# is dense in \‚] . In particular, this shows that the product of two separable spaces is separable.

Example 5.12 The open intervals Ð+ß ,Ñ form a base for the usual topology in ‘ , so each set Ð+ß,Ñ‚Ð-ß.Ñ is in the base U for the product topology on ‘‚ ‘ . It is easy to see that every “open box” Y‚Z in U can be written as a union of “simple open boxes” like Ð+ß,Ñ ‚ Ð-ß.Ñ . Therefore Uw œ ÖÐ+ß,Ñ‚Ð-ß.ÑÀ + @ ,ß - @ .× also is a (simpler) base for the product topology on ‘‘‚ . From this, it is clear that the product topology on ‘‚ ‘ is the usual topology on the plane ‘# (see Example 5.10 ).

In general, the open sets YZ and in the base for the product topology on \ ‚ ] can be replaced by sets “Y chosen from a base for \Z ” and “ chosen from a base for ] ,” as in this example. So in the definition of the product topology, it is sufficient to say that basic open sets are of the form Y‚Zß where Y+8. Vare basic open sets from \ and from ]Þ

Definition 5.13 We say that a space Ð\ßg Ñ satisfies the second axiom of countability (or, more simply, that Ð\ßg Ñ is a second countable space) if it is possible to find a countable base U for the topology g .

For example, ‘ is second countable because, for example, UœÖÐ+ß,ÑÀ+ß,− × is a countable base. Is ‘# is second countable (why or why not ) ?

" Example 5.14 The collection UœÖÒBßB18 ÑÀB− ‘ ß8− × is a base for the Sorgenfrey " topology on ‘ . But the collection ÖÒBßB18 ÑÀB− ß8− × is not a countable base for the Sorgenfrey topology. Why not?

Since the sets in a base may be simpler than arbitrary open sets, they are often more convenient to work with, and working with the basic open sets is often all that is necessary
not a surprise since all the information about the open sets in contained in the base U . For example, you should check that

1) If U is a base for g , then B−E cl iff each basic open set F containing B satisfies F∩E Á gÞ

w 2) If 0 ÀÐ\ß.ÑÄÐ]ß. Ñ and U is a base for the topology g . w on ] , then 0 is continuous iff 0ÒFÓ
" is open for each F− U . This means that we needn't check the inverse images of all open sets to verify that 0 is continuous.

118 C. Subbase for the topology

Definition 5.15 Suppose Ð\ßg Ñ is a topological space. A family Æ of open sets is called a subbase for the topology g if the collection U of all finite intersections of sets from Æ is a base for g . (Clearly, if UgU is a base for , then is automatically a subbase for g .)

Examples i) The collection Æ œÖMÀ MœÐ
∞ß,Ñ or MœÐ+ß∞Ñß+ß,−‘ × is a subbase for a topology on ‘ . All intervals of the form Ð+ß,Ñ œ Ð
∞ß,Ñ∩Ð+ß∞Ñ are in U, so U is a base for the usual topology on ‘ .

ii) The collection Æ of all sets Y‚] and Z‚\ÐY for open in \ and Z open in ] ) is a subbase for the product topology on \‚] : these sets are open in \‚] and the collection U of all finite intersections of sets in Æ includes all the open boxes Y ‚Z œÐY ‚]Ñ∩Ð\‚ZÑ .

We can define a topology on a set \ by giving a collection Æ of subsets as the subbase for a topology. Surprisingly, any collection Æ can be used: no special conditions on Æ are required.

Theorem 5.16 (The Subbase Theorem) Suppose \ is a set and Æ is any collection of subsets of \ . Let U be the collection of all finite intersections of sets from Æ . Then U is a base for a topology g , and Æ is a subbase for g .

Proof First we show that U satisfies conditions 1) and 2) of The Base Theorem.

1) \ −U since \ is the intersection of the empty subcollection of Æ ( this follows the convention that the intersection of an empty family of subsets of \ is \ itself. See Example I.4.5.5 ). Since U certainly { : U } \−ß \œ F F −Þ 2) Suppose F"# and F−U , and B−F∩F "# . We know that Fœ∩∩ ""7S ... S and F∩∩# œ S71" ... S 715 for some S,...,S,...,S " 7 715 − Æ , so

B−FœF∩F$ " # œW∩ " ... ∩W 7 ∩W 71" ∩ ... ∩W 715 − U

Therefore gœÖSÀS is a union of sets from U × is a topology, and Ug is a base for . By definition of Ug and , we have ÆUg© © and each set in g is a union of finite intersections of sets from Æ . Therefore Æ is a subbase for g . ñ

Example 5.17

1) Let „œÖ#ß%ß'ßÞÞÞש Æ and œÖÖ"×ßÖ#×ß „Æ ×Þ is a subbase for a topology on . A base for this topology is the collection U of all finite intersections of sets in Æ :

UœÖg , ßÖ#×ßÖ"×ß „ × .

The not-very-interesting topology g generated by the base U is collection of all possible unions of sets from U : gœÖg , , Ö" }, {2}, {1,2}, „„, ∪Ö"××

119 2) For each 8− ß let W8 œÖ8ß81"ß81#ßÞÞÞ×Þ The collection Æ œÖW8 À8− × is a subbase for a topology on . Here, W∩W8 7 œW 5 where 5œ max Ö7ß8× , so the collection of all finite intersections from Æ is just Æ itself. So UÆœ is actually a base for a topology. The topology is gœ Æ ∪Ög× .

3) Let ÆœÖjÀj is a straight line in ‘‘# × in # . For every point :−Ö:× ‘ # , is the intersection of two sets from ÆUÆ , so Ö:×− Þ is a subbase for the discrete topology on ‘ # .

4) Let ÆœÖjÀj is a vertical line in ‘Æ# ×Þ generates a topology on ‘ # for which U‘œÖßg∪# } Æ is a base and g œÖSÀS is a union of vertical lines ×Þ

5) Let Æ be a collection of subsets of \ and suppose that g w is any topology on \ for which Æ© gw Þ Since g w is a topology, it must contain all finite intersections of sets in Æ , and therefore must contain all possible unions of such intersections. Therefore g w contains the topology g for which Æ is a subbase. To put it another way, the topology for which Æ is a subbase is in the smallest topology on \ containing the collection Æ. In fact, as an exercise, you can check that

g ggw w Æg and w is a topology on . œÖÀª \×

Caution We said earlier that for some purposes it is sufficient (and simpler) to work with basic open sets, rather than arbitrary open sets
for example to check whether B−E cl , it is sufficient to check whether Y∩EÁg for every basic open set Y that contains B . However, it is not always sufficient to work with subbasic open sets. Some caution is necessary. For example, Æ œÖMÀ MœÐ
∞ß,Ñ or MœÐ+ß∞Ñß+ß,−‘ × is a subbase for the " usual topology on ‘ . We have M∩Ág ™ for every subbasic open set M containing # , but " # Âcl ™ Þ

D. The closure operator

Usually we describe a topology g by giving a subbase ÆU, a base , or by giving collections U B to be the basic neighborhoods ateach point B . In the early history of general topology, one other method was sometimes used. We will never actually use it, but we include it here as a curiosity.

Let clg be the closure operator in Ð\ßg Ñ ( normally, we would just write “cl” for the closure operator; here we write “clg ” to emphasize that this closure operator comes from the topology g on \). It gives us all the information about g . That is, using clg ß we can decide whether any set E is closed (by asking “is clg E œ E ?”) and therefore can decide whether any set F is open (by asking “is \
F closed ?”). It should be not be a surprise, then, that we can define a topology on a set \ if we are start with an “operator” which “behaves like a closure operator.” How is that? Our first theorem tells us the crucial properties of a closure operator.

120 Theorem 5.18 Suppose Ð\ßÑg is a topological space and EßF are subsets of \ . Then

1) cl g g œ g 2) E © cl g E 3) E is closed iff EœE cl g 4)cl g E œ clgÐ cl g E Ñ 5) clgÐE∪B Ñ œ cl g E∪ cl g F

Proof 1) Since g is closed, cl g gœg

2) is closed and cl E© ÖJÀJ E©Jלg E 3) ÊE : If is closed, then E itself is one of the closed sets J used in the definition clg Eœ∩ÖJÀJ is closed and JªE×ß so Eœ cl g E . ÉEœEE : If clg , then is closed because clg E is an intersection of closed sets.

4) clgE is closed, so by 3), cl ggg EÐEÑœ cl cl

5) EFE∪ª , so clg ÐE ∪FÑ ªE cl g. Similarly, cl g ÐE ∪FÑ ªF cl g . Therefore clgÐE∪Ñ Fª cl g ÐE∪ Ñ cl g ÐF Ñ . On the other hand, clgE ∪ cl g F is a closed set that contains E F∪ , and therefore clgE∪ cl g Fª cl g ÐE∪ F Ñ . ñ

The next theorem tells us that we can use an operator “cl” to create a topology on a set.

Theorem 5.19 (The Closure Operator Theorem) Suppose \ is a set and that for each E © \, a subset cl E is defined ( that is, we have a function cl À cÐ\ÑÄ c Ð\Ñ ) in such a way that conditions 1), 2), 4) and 5) of Theorem 5.18 are satisfied. Define g œ { S À clÐ\
Sќ\
S }. Then g is a topology on \ , and cl is the closure operator for this topology (that is, clœ cl g ).

Note: i) Such a function cl is called a “Kuratowski closure operator,” or just “closure operator” for short . ii) The Closure Operator Theorem does not ask that cl satisfy condition 3): initially, there is no topology on the given set \ , so 3) would be meaningless. But 3) motivates the definition of g as the collection of sets whose complements are unchanged when “cl” is applied..

Proof (Numbers in parentheses refer to properties of “cl” )

First note that: (5) (*) if E©F , then clE©E∪FœÐE cl cl cl ∪FÑœ cl F g has the properties required for a topology on the set \ À Ð#Ñ i) \© cl \ , so \œ cl \Þ Therefore cl Ð\
gÑœ cl \œ\œ\
g , so g− g . (1) Also, clÐ\
\Ñœ cl gœgœ\
\ , so therefore \−g Þ

121 ii) Suppose Oα −g for each α −E . For each particular α ! −E , we have (*) clÐ\
O Ñ©Ð\
cl O ќ\
O because O − g . This is true for every α α! α ! α ! α A , so clO O ) O . ! − Ð\
α Ñ©Ð\
α œ\
α But by 2), we know thatO cl O . \
α ©Ð\
α Ñ Therefore clO O , so O g . Ð\
α Ñœ\
α α −

iii) If OO"# and are in g , then clÐ\
( OO" ∩# ÑÑœ cl ÐÐ\
OO " Ñ∪Ð\
# ÑÑ (5) œÐ\ cl
O" Ñ∪Ð\ cl
O # ÑœÐ\
O " Ñ∪Ð\
O # Ñ (since OO "# and − g ) œ\
(OO" ∩# ). Therefore OO " ∩ # −g . Therefore g is a topology on \ .

Having this topology g now gives us an associated closure operator clg and we want to show that clg œ cl. First, we observe that for closed sets in Ð\ßg ÑÀ

(**) clg FœF iff F is closed in Ð\ßÑg iff \
F−g iff cl Ð\
Ð\
FÑÑ œ \
Ð\
FÑ iff cl FœF .

Tofinish, we must show that cl Eœ clg E for every E©\ .

E©E clg , so (*) gives that cl E©ÐE clclgÑ . But cl g E is closed in Ð\ßÑg , so usingFœE clg in (**)gives clcl gg EœE cl . Therefore cl E©Ecl g . (4) On the other hand, clclEœ cl E , so using FœE cl in (**) gives that clE is closed in Ð\ßÑÞg (2) But clEªE , so cl E is one of the closed sets in the intersection that defines clgE . Therefore cl EªEÞ cl g Therefore cl EœEñ cl g .

Example 5.20 Eif E is finite 1) Let \ be a set. For E © \ , define cl E œ . \if E is infinite Then cl satisfies the conditions in the Closure Operator Theorem. Since clEœE iff E is finite or E œ \ , the closed sets in the topology generated by cl are precisely \ and the finite sets
that is, cl generates the cofinite topology on \ .

2) For each subset E of ‘ , define

clEœÖB−À‘ there is asequence Ð+ÑE8 in with each + B 8 and | +
BlÄ!× 8 .

It is easy to check that cl satisfies the hypotheses of the Closure Operator Theorem. Moreover, a set E is open in the corresponding topology iff aB−Eb,B such that ÒBß,Ñ©E . Therefore the topology generated by cl is the Sorgenfrey topology on ‘ . What happens in this example if “ ” is replaced by “  ” in the definition of cl ?

Since closures, interiors and Frontiers are all related, it shouldn't be surprising that we can also describe a topology by defining an appropriate “int” operator or “Fr” operator on a set \ .

122 E1. Let \œÖ!ß"ß#ßÞßÞßÞלÖ!×∪ . For S©\ , let

9SÐ8Ñœ the number of elements in S∩Ò"ß8ÓœlS∩"ß8ÓlÞ [ Then define g œÖSÀ!ÂS or Ð!−S and lim9SÐ8Ñ œ 1) × . 8 Ä ∞ 8

a) Prove that g is a topology on \ .

b) In anytopological space: a point B is called a limit point of the set E if R∩ÐE
ÖB×ÑÁg for every neighborhood R of B . Informally, B is a limit point of E means that there are points of Eß other than Bß itself arbitrarily close to B . Prove that in any topological space, a set F is closed iff F contains all of its limit points.

c) For Ð\ßÑg as defined in a): prove that B is a limit point of \ if and only if Bœ! .

E2. Suppose Ð\ßÑg is a topological space and that E©\α for each α −E .

a) Prove that if clα is closed, then Ö Eα À −E× Ö cl EÀα −Eל cl ÖEÀ α −E× α α

(Note that “© ” is true for any collection of sets E α . )

b) A family ÖFÀ−E×α α of subsets of \ is called locally finite if each point B−\ has a neighborhood R such that R∩FÁgα for only finitely many α 's. Prove that if ÖFα À α −E× is locally finite, then

Ö cl FÀα −Eל cl ÖFÀ α −E× α α c) Prove that in Ð\ßg Ñ , the union of a locally finite family of closed sets is closed.

E3. Suppose 0 À Ð\ß.Ñ Ä Ð]ß=ÑÞ Let U be a base for the topology g= and let Æ be a
" subbase for g=Þ Prove or disprove: 0 is continuous iff 0ÒSÓ is open for all S− U iff 0ÒSÓ
" is open for all S−ÞÆ

E4. A space Ð\ßÑg is called a X"-space if ÖB× is closed for every B−\Þ

a) Give an example where Ð\ßÑg is not a X " -space and g is not the trivial topology. b) Prove that \X is a " -space if and only if, given any two distinct points Bß C − \ , each point is contained in an open set not containing the other point. c) Prove that in a X" -space, each set ÖB× can be written as an intersection of open sets. d) Prove that a subspace of a X" -space is a X " -space. e) Prove that if a pseudometric space Ð\ß.Ñ is a X" -space, then . must in fact be a metric. f) Prove that if \ and ] are X" -spaces, so is \‚]Þ

123 E5. Ð\ßg Ñ is called a X#-space ( or Hausdorff spac e ) if whenever BßC−\ and BÁC , then there exist disjoint open sets Y and Z with B−Y and C−Z .

a) Give an example of a space Ð\ßÑg which is a X" -space but not a X # -space (see E4. ) . b) Prove that a subspace of a Hausdorff space is Hausdorff. c) Prove that if \] and are Hausdorff, then so is \‚] .

E6. Prove that every infinite X# -space contains an infinite discrete subspace
that is, a subset which is discrete in the subspace topology (see E5 ).

E7. Suppose that Ð\ßg Ñ and Ð]ß g w Ñ are topological spaces. Recall that the product topology on \ ‚ ] is the topology for which a base is the collection of “open boxes”

UœÖY‚ZÀY− g ßZ − g w ×

Therefore a set S © \ ‚ ] is open in the product topology iff for all ÐBß CÑ − S , there exist open sets Y in \ and Z in ] such that ÐBßCÑ−Y‚Z©SÞ ( Note that the product topology on ‘‚ ‘ is the usual topology on ‘ # . (We always assume that the topology used on a product of two spaces \ ‚ ] is the product topology unless something different is explicitly stated. )

a) Verify that U is, in fact, a base for a topology on \ ‚ ] . b) Consider the projection map 1" À\‚]Ä\ . Prove that if S is any open set in \‚]Ð not necessarily a “box” Ñ , then 1" ÒSÓ is open in \ . ( We say 1" is an open map . The same is true for the projection 1#Þ ) c) Prove that if E©\ and F©] , then cl\‚] ÐE‚FÑœ cl \ E‚ cl ] FÞ Use this to explain why “the product of two closed sets is closed in \ ‚ ] .” d) Show that \‚] has a countable base iff each of \] and has a countable base. e) Show that there is a countable neighborhood base at ÐBßCÑ − \ ‚ ] iff there is a countable neighborhood base UB at B − \ and a countable neighborhood base U C at C −]Þ f) Suppose Ð\ß ." Ñ and Ð] ß . # Ñ are pseudometric spaces. Define a pseudometric . on the set \ ‚ ] by

.ÐÐB"" ßC ÑßÐB ## ßC ÑÑœ. ""# ÐB ßB Ñ1. #"# ÐC ßC ÑÞ

Prove that the product topology on \ ‚ ] is the same as the topology g. .

Note : . is the analogue of the taxicab metric in ‘#Þ There are other equivalent pseudometrics that produce the product topology on \ ‚ ] , for example

w # #"Î# . ÐÐBßCÑßÐBßCÑÑ"" ## œÐ.ÐBßBÑ ""# 1.ÐCßCÑÑ #"# ß and

ww . ÐÐBßCÑßÐBßCÑÑœ"" ## max Ö.ÐBßBÑß.ÐCßCÑ×Þ ""##"#

124 E8. Suppose E©Ð\ßÑÞg The set E is called regular open if EœÐEÑ int cl and E is regular closed if E œ cl(int E ).

a) Show that for any subset F

i) \
cl F œ int Ð\
FÑ ii) \
int F œ cl Ð\
FÑ

b) Give an example of a closed subset of ‘ which is not regular closed.

c) Show that the complement of a regular open set in Ð\ßg Ñ is regular closed and vice-versa.

d) Show that the interior of any closed set in Ð\ßg Ñ is regular open.

e) Show that the intersection of two regular open sets in Ð\ßg Ñ is regular open.

f) Give an example of the union of two regular open sets that is not regular open.

E9. In each part, prove the statement or provide a counterexample:

a) For any B in a topological space Ð\ßÑßÖB×g is equal to the intersection of all open sets containing B .

b) In a topological space, a finite set must always be closed.

c) Suppose we have topologies g on , one for each α . Then { gα : } is α \ −E α −E also a topology on \ .

d) Ifg"# and g are topologies on X , then there is a unique smallest topology g $ on X such that g"∪ g # © g $ .

e) Suppose, for each α− E , that g α is a topology on \ . Then there is a unique smallest topology g on \ such that for each αgg , ªα Þ

E10. Assume that natural number, except " , can be factored into primes; you shouldn't need any other information about prime numbers. For +−™ and .− , let

F+ß. œÖÞÞÞß+
#.ß+
.ß+ß+1.ß+1#.ßÞÞÞלÖ+15.À5−™ × and let UœÖF+ß. À+− ™ ß.− ×

a) Prove that U is a base for a topology g on ™ b) Show that each set F+ß. is closed in ÐßÑ™ g c) What is the set is a prime number ? ÖFÀ:!ß: × d) Part c) tells you what famous fact about the set of prime numbers?

125 6. Countability Properties of Spaces

Countable sets are often easier to work with than uncountable sets, so it is not surprising that spaces with certain “countability properties” are viewed as desirable. Most of these properties have already been defined, but the definitions are collected together here for convenience.

Definition 6.1 Ð\ßg Ñ is called

first countable if we can choose a countable neighborhood base UB at every point B − \ . (We also say that \ satisfies the first axiom of countability . )

second countable if there is a countable base U for the topology g . ( We also say that \ satisfies the second axiom of countability . )

separable if there is a countable dense set H in \

Lindelo¨f if whenever h is a collection of open sets for which h , then there is œ \ a countable subcollection hw h for which h w œÖYßYßÞÞÞßYßÞÞÞש" # 8 œ\Þ h is called an open cover of \ , and hw iscalled a subcover from h . Thus, \ is Lindelo¨f if “every open cover has a countable subcover .”)

Example 6.2

1) A countable discrete spaceÐ\ßÑg is second countable because U œÖÖB×ÀB−\× is a countable base. Is \ first countable? separable? Lindelo¨f ?

2) ‘ is second countable because UœÖÐ+ß,ÑÀ+ß,− × is a countable base. Similarly, 8 ‘ is second countable since the collection U of boxes ÐB"" ßC Ñ‚ÐB ## ßC Ñ‚ÞÞÞ‚ÐB 88 ßC Ñ with rational endpoints is a countable base (check! )

3) Let \ be a countable set with the cofinite topology g . \ has only countably many finite subsets (see Theorem I.11.1 ) so there are only countably many sets in g . Ð\ß g Ñ is second countable because we could choose Uœ g as a countable base.

The following theorem implies that each of the spaces in Example 6.2 is also first countable, separable, and Lindelo¨f. ( However, it is really worthwhile to try to verify these properties, in each example, directly from the definitions. )

Theorem 6.3 A second countable topological space (\ßg Ñ is also separable, first countable, and Lindelo¨f.

Proof Let U œ ÖS" ßS # ßÞÞÞS 8 ßÞÞÞ× be a countable base for g .

i) For each 8 , pick a point B−S8 8 and let HœÖBÀ8−× 8 . The countable set H is dense. To see this, notice that if Y is any nonempty open set in \ , then for some 8 , B−S©Y8 8 so Y∩HÁg . Therefore \ is separable.

126 ii) For each B−\ß let UB œS−ÀB−S× { U . Clearly U B is a neighborhood base at B , so \ is first countable.

iii) Let h be any open cover of \B−\ . If , then B− some set Y−ÞB h For each B , we can then pick a basic open set S−BU such that B−S©Y BB . Let i œÖSÀB−\×Þ B Since each SB −U ß there can be only countably many different sets S B : that is, i may contain “repeats.” Eliminate any “repeats” and list only the different sets in i , so i œÖSßSßÞÞÞSßÞÞÞ×BBBBB" # 8where S©Y−Þ 8 8 h Every B is in one of the sets S B 8 , so w h œÖYB" ßY B # ßÞÞÞY B 8 ßÞÞÞ× is a countable subcover from h . Therefore \ is Lindelo¨f. ñ

The following examples show that no other implications exist among the countability properties in Theorem 6.3.

Example 6.4

1) Suppose \ is uncountable and let g be the cofinite topology on \ .

Ð\ßg Ñ is separable since any infinite set is dense.

Ð\ßÑg is Lindelo¨f. To see this ß let h be an open cover of \ . Pick any one nonempty setY−h . Then \
Y is finite, say \
YœÖBßÞÞÞßB×Þ" 8 For w each Bß3 pick a set Y− 333h with B−YÞ Then h œÖY×∪ÖYÀ3œ"ßÞÞÞß8× 3 is a countable (actually, finite ) subcover chosen from h.

However \ is not first countable (Example 4.5.3), and therefore, by Theorem 6.3, \ is also not second countable.

2) Suppose \ is uncountable. Define g œÖS©\ÀSœg\
S or is countable × .

g is a topology on \ (check! ) called the cocountable topology. A set G © \ is closed iff Gœ\ or G is countable. ( This is an “upscale” analogue of the cofinite topology. )

An argument very similar to the one in the preceding example shows that Ð\ßg Ñ is Lindelo¨f. But Ð\ßÑg is not separable
every countable subset is closed andtherefore not dense. By Theorem 6.3, Ð\ßg Ñ also cannot be second countable.

3) Suppose \ is uncountable set and choose a particular point : − \Þ Define g œÖS©\ÀSœg or :−S× . (Check that g is a topology. )

Ð\ßÑg is separable because Ö:× is dense.

Ð\ßÑg is not Lindelo¨f
because the cover h œÖÖBß:×ÀB−\× has no countable subcover.

Is Ð\ßg Ñ first countable?

127 4) Suppose \ is uncountable and let g be the discrete topology on \ . Then Ð\ßÑg is

first countable because UB œÖÖB×× is a neighborhood base at B not second countable because each open set ÖB× would have to be in a base U not separable because clHœHÁ\ for any countable set H not Lindelo¨f because the cover h œ ÖÖB× À B − \× has no countable subcover. ( In fact, not a single set can be omitted from h : h has no propersubcover ÞÑ

For “special” topological spaces
pseudometrizable ones, for example
it turns out that things are better behaved. For example, we noted earlier that every pseudometric space Ð\ß .Ñ is first countable (Example 4.3 ). The following theorem shows that in Ð\ß .Ñ the other three countability properties are equivalent to each other: that is, either all of them are true in Ð\ß .Ñ or none are true.

Theorem 6.5 Any pseudometric space Ð\ß.Ñ is first countable. Ð\ß.Ñ is second countable iff Ð\ß.Ñ is separable iff Ð\ß.Ñ is Lindelo¨f.

Proof i) Second countableÊ Lindelo¨f: by Theorem 6.3, this implication is true in any topological space.

ii) Lindelo¨fÊ separable: suppose Ð\ß.Ñ is Lindelo¨f. For each 8− , let h 8 œÖFÐBÑÀ8−" ßB−\×. For each 8 , h8 is an open cover so h 8 has a countable 8 " subcover
that is, for each 8 we can find countably many 8 -balls that cover \ À say ∞ \œ FÐBÑÞ" 8ß5 Let H be the set of centers of all these balls: HœÖBÀ8ß5−×Þ8ß5 For 5œ" 8 " any B−\ and every 8 , we have B−FÐBÑ" 8ß5 for some 5 , so .ÐBßBÑ@ 8ß5 . Therefore, for 8 8 every 8FÐBÑ∩HÁg
, " in other words, B can be approximated arbitrarily closely by points 8 from H . Therefore H is dense, so Ð\ß.Ñ is separable.

iii) SeparableÊ second countable: suppose Ð\ß.Ñ is separable and that HœÖBßBßÞÞÞßBßÞÞÞ×" # 5 is a countable dense set. Let U œÖFÐBÑÀ8ß5−" 5 × . U is a 8 countable collection of open balls and we claim U is a base for the topology g .Þ

Suppose C−S−X. Þ By the definition of “open,” there is an % > 0 for which " % " FÐCÑ©S% . Pick 8 so that 8 @ß # and pick B−H5 so that .ÐBßCÑ@Þ 5 8 Then C−FÐBÑ©FÐCÑ©S" 5 % (because D−FÐBÑ" 5 8 8 " " % Ê.ÐDßCÑŸ.ÐDßBÑ1.ÐBßCÑ@5 5 8 1 8 @#Ð # Ñœ% ). ñ

It's customary to call a metric space that has these three equivalent properties a “separable metric space” rather than a “second countable metric space” or “Lindelof¨ metric space.”

Theorem 6.5 implies that the spaces in parts 1), 2), 3) of Example 6.4 are not pseudometrizable. In general, to show a space Ð\ßg Ñ is not pseudometrizable we can i) show that it fails to have some property shared by all pseudometric spaces (for example, first countability), or ii) show that it has one but not all of the properties “second countable,” “Lindelo¨f,” or “separable.”

128 Exercises

E11. Define g œÖY∪ZÀY is open in the usual topology on ‘ and Z©× .

a) Show that g is a topology on ‘ . If B is irrational, describe an “efficient” neighborhood base at BB . Do the same if is rational.

b) Is Б ß g Ñ first countable? second countable? Lindelo¨f ? separable? The space Б ß g Ñ is called the “scattered line.” We could change the definition of g by replacing with some other set E © ‘ . This creates a space in which the set E is “scattered.” ∞ Hint: See Example I.7.9.6. It is possible to find open intervals M8 such that 8œ" M 8 ª ∞ and for which lengthÐM8 Ñ @ " . 8œ"

E12. A point B − Ð\ßg Ñ is called a condensation point if every neighborhood of B is uncountable.

a) Let G be the set of all condensation points in \ÞG Prove that is closed.

b) Prove that if \ is second countable, then \
G is countable.

E13. Suppose Ð\ßg Ñ is a second countable space and let U be a countable base for the topology. Suppose U w is another base (not necessarily countable ) for g containing open sets all of which have some property T . (For example, “ T ” could be “clopen” or “separable.”) Show that there is a countable base U ww consisting of open sets with property T Þ Hint: think about the Lindelof¨ property.

E14. A space Ð\ßg Ñ is called hereditarily Lindelof¨ if every subspace of \ is Lindelo¨f Þ

a) Prove that a second countable space is hereditarily Lindelo¨f.

In any space, a point B is called a limit point of the set E if R∩ÐE
ÖB×ÑÁg for every neighborhood RBÞ of Informally, B is a limit point of E if there are points in E different from B and arbitrarily close to B . )

b) Suppose \ is hereditarily Lindelo¨f Þ Prove that the set EœÖB−\ÀB is not a limit point of \× is countable.

129 E15. A space Ð\ßg Ñ is said to satisfy the countable chain condition (œ CCC Ñ if every family of disjoint open sets must be countable.

a) Prove that a separable space Ð\ßg Ñ satisfies the CCC.

b) Give an example of a space that satisfies the CCC but that is not separable. ( It is not necessary to do so, but can you find an example which is a metric space?)

E16. Suppose Ð\ßg Ñ is a topological space and that c and U are two bases for the topology g, and that c and U are infinite.

a) Prove that there is a subfamily UUw© such that U w is also a base and llŸll Uc w . (Hint: For each pair >œÐYßZÑ−‚c c , pick, if possible, a set [−> U such that Y©[©Z; otherwise set [œgÞ> )

b) Use part a) to prove that the Sorgenfrey line is not second countable. (Hint: Show that otherwise there would be a countable base of sets of the form Ò+ß ,Ñß but that this is impossible. Ñ

130 7. More About Subspaces

Suppose Ð\ßÑg is a topological space Þ In Definition 3.1, we defined the subspace topology g E on E©\À gE œÖE∩SÀS− g × . In this section we explore some simple but important properties of subspaces.

If E © F © \ , there are two ways to put a topology on E À

1) we can give E the subspace topology gE from \ , or

2) we can give F the subspace topology gF , and then give E the subspace topology from the space ÐFßÑ
gF that is, we can give E the topology ÐÑÞg F E

In other words, we can think of E as a subspace of \ or as a subspace of the subspace F . Fortunately, the next theorem says that these two topologies are the same. More informally, Theorem 7.1 says that in a space \ “a subspace of a subspace is a subspace.”

Theorem 7.1 If E©F©\ , and g is a topology on \ , then gE œ ( g F ) E .

Proof Y−gE iff YœS∩E for some S− g iff YœS∩ÐF∩EÑ iff YœÐS∩FÑ∩E . But S∩F− gF , so the last equation holds iff Y−ÐÑñ g F E .

We always assume a subset E has the subspace topology (unless something else is explicitly stated). The notation E©Ð\ßÑg emphasizes that E is considered a subspace , not merely a subset Þ

By definition, a set is open in the subspace topology on E iff it is the intersection with E of an open set in \ . The same is also true for closed sets.

Theorem 7.2 Suppose J©E©Ð\ßÑJg . is closed in E iff JœE∩G where G is closed in \ .

Proof J is closed in EE
J iff is open in EE
JœS∩E iff (for some open set S\ in ) iff JœE
ÐS∩EÑœÐ\
SÑ∩EœG∩E(where Gœ\
S is a closed set in \ ). ñ

Theorem 7.3 Suppose E©Ð\ßg Ñ .

1) Let a − E . If U+ is a neighborhood base at a in \ , then ÖF∩EÀF−U + × is a neighborhood base at + in EÞ

2) If U is a base for g , then ÖF∩EÀF −U × is a base for g E Þ

With a slight abuse of notation, we can informally write these collections as U+ ∩E and. U ∩EÞ Why is this an “abuse?” What do U+ ∩ E and U ∩ E mean if taken literally ?

131 Proof 1) Suppose +−E and that R is a neighborhood of +E in . Then +−R©R intE , so there is an open set S in \ such that +− int E RœS∩E©SÞ Since U+ is a neighborhood base at +\ in , there is a neighborhood F− U + such that +−F©SÞ Then +−Ð int\ FÑ∩E©F∩E−U + ∩E . Since Ðint \ FÑ∩E is open in E , we see that F∩E is a neighborhood of +E in . And since +−F∩E©S∩Eœ intE R©R , we see that U+ ∩E is a neighborhood base at +Eñ in .

2) Exercise

Theorem 7.3 tells us that in the subspace E we can get a neighborhood base at a point + by choosing a neighborhood base at + in \ are then restricting all its sets to E ; and that the same applies to a base for the subspace topology.

Corollary 7.4 Every subspace of a first countable (or second countable) space Ð\ßg Ñ is first countable (or second countable).

" # "# Example 7.5 Suppose W is a circle in ‘ and that :−W©Þ ‘U : œÖFÐ:ÑÀ!×% % is a # " neighborhood base at : in ‘ , and therefore U : ∩W is a neighborhood base at : in the subspace " " " W. The sets in U: ∩W are “open arcs on W containing : .” (See the figure,)

The following theorem relates closure in a subspace to closure in the larger space. It turns out to be a very useful technical observation.

Theorem 7.6 Suppose E©F©Ð\ßg Ñ , then clF EœF∩EÞ cl \

Proof F∩E cl\ is a closed set in F that contains EßF∩EªEÞ so cl\ cl F

On the other hand, suppose ,−F∩E cl\ . To show that ,−E cl F , pick an open set Y in F that contains , . We need to show Y∩EÁg . There is an open set S\ in such that S∩FœY. Since ,− cl\ E , we have that gÁS∩EœS∩ÐF∩EÑœÐS∩FÑ∩E œY∩E. ñ

132 Example 7.7 1) œ cl œ cl ‘ ∩

2) clÐ!ß#Ó Ð!ß"Ñ œ cl ‘ Ð!ß"Ñ∩Ð!ß#Ó œ Ò!ß"Ó∩Ð!ß#Ó œ Ð!ß"Ó

3) The analogous calculations are not necessarily not true for interiors and boundaries . For example:

œ int Á int ‘ ∩œg , and

gœ Fr Á Fr ‘ ∩œÞ

Why does “cl” have a privileged position here? Is there a “reason” why you would expect a better connection between closures in E and closures in \ than you would expect between interiors in E and interiors in \ ?

Definition 7.8 A property T of topological spaces is called hereditary if whenever a space \ has property T , then every subspace E also has property T .

For example, Corollary 7.4 tells us that first and second countability are hereditary properties. Other hereditary properties include “finite cardinality” and “pseudometrizability.” On the other hand, “infinite cardinality” is not a hereditary property.

Example 7.9

1) Separability is not a hereditary property. For example, consider the Sorgenfrey plane \Ðsee Example 5.4 ) Þ\ is separablebecause # is dense. Consider the subspace HœÖÐBßCÑÀB1Cœ"× . The set YœÒ+ß+1"Ñ‚Ò,ß,1"Ñ is open in \ so if Ð+ß,Ñ−Hß then Y∩HœÖÐ+ß,Ñ× is open in the subspace HÞ Therefore H is a discrete subspace of \ , and an uncountable discrete space is not separable. Similarly, the Moore place > is separable (see Example 5.6 ); the B -axis in > is an uncountable discrete subspace which is not separable.

2) The Lindelo¨f property is not heredity. Let E be an uncountable set and let \œE∪Ö:×, where : is any additional point not in E . Put a topology on \ by giving a neighborhood base at each point. U œÖÖ+×× for +−E + U: œÖFÀ:−Fand \
F is finite ×

(Check that the collections UB satisfy the hypotheses of the Neighborhood Base Theorem 5.2. )

If i is an open cover of \:−Z , then for some Z−i . By ii), every neighborhood of :\ in has a finite complement, so \
Z is finite. For each C in the finite set \
Z , we can choose a w set Z−Ci with C−ZÞ C Then i œÖZ×∪ÖZÀC−\
Z× C is a countable (in fact, finite) subcover from i , so \ is Lindelo¨f.

The definition of U+ implies that each point of E is isolated in E ; that is, E is an uncountable discrete subspace. Then h œÖÖ+×À+−E× is an open cover of E that has no countable subcover. Therefore the subspace E is not Lindelo¨f.

133 Even when a property is not hereditary, it is sometimes “inherited” by certain subspaces
perhaps, for example, by closed subspaces, or by open subspaces. The next theorem illustrates this.

Theorem 7.10 A closed subspace of a Lindelo¨f space is Lindelo¨f ( so we say that the Lindelof¨ property is “closed hereditary” ).

Proof Suppose O is a closed subspace of the Lindelo¨f space \ . Let h œÖYÀα α −E× be a cover of OY by sets α that are open in O . For each α , there is an open set Zα in \ such that Zα ∩OœY α Þ

Since O is closed, \
O is open and i œÖÖ\
O××∪ÖZÀ−E×α α is an open cover of \Þ w But \ is Lindelo¨f, so i has a countable subcover from i , say i œ Ö\
OßZ" ßÞÞÞßZ 8 ßÞÞÞ×Þ (The set \
O might not be needed in i w , but it can't hurt to include it. ) Clearly then, the collection ÖYßÞÞÞßYßÞÞÞ×" 8 is a countable subcover of O from h . ñ

Note:

1. A little reflection on the proof shows that to prove O is Lindelof,¨ it would be equivalent to show that every cover of O by sets open in \ has a countable subcover ÞÑ

2. A space \ with the property that every open cover has a a finite subcover is called compact . See, for example, the space in Example 6.4.1. An obvious “tweak” to the proof for Theorem 7.10 shows that a closed subspace of a compact space is compact. We will look at the properties of compact spaces in much more detail in Chapter 4 and beyond.

134 8. Continuity

We first defined continuous functions between pseudometric spaces by using the distance function . to mimic the definition of continuity as given in calculus. But then we saw that our definition could be restated in other equivalent ways in terms of open sets, closed sets, or neighborhoods. For topological spaces, we do not have distance functions available to use in a definition of continuity. But we can still make a definition using neighborhoods (or open sets, or closed sets) since the neighborhoods of + describe “nearness” to + . Of course, the definition parallels the way neighborhoods describe continuity in pseudometric spaces.

Definition 8.1 A function 0ÀÐ\ßÑÄÐ]ßÑg g w is continuous at + − \ if whenever R is a neighborhood of 0Ð+Ñ , then 0ÒRÓ
" is a neighborhood of + . We say 0 is continuous if 0 is continuous at each point of \ .

The statement that 0 is continuous at + is clearly equivalent to each of the following statements: :

i) for each neighborhood R0Ð+Ñ of there is a neighborhood [+ of such that 0Ò[ Ó © R ii) for each open set Z containing 0Ð+Ñ there is an open set Y containing + such that 0ÒYÓ © Z iii) for each basic open set Z containing 0Ð+Ñ there is a basic open set Y containing + such that 0ÒYÓ©Z .

In the following theorem, the conditions i) -iii) for continuity are the same as those in Theorem II.5.6 for pseudometric spaces. Condition iv) was not mentioned in Chapter II, but it is sometimes handy.

Theorem 8.2 Suppose 0ÀÐ\ßg ÑÄÐ]ß g w Ñ . The following are equivalent.

i)0 is continuous ii)if S−gw ß then 0
" ÒSÓ− g (the inverse image of an open set is open ) iii) if J is closed in ] , then 0ÒJÓ
" is closed in \ ( the inverse image of a closedset isclosed ) iv) for every E©\À 0Ò cl\ ÐEÑÓ© cl ] Ð0ÒEÓÑ .

Proof The proof that i)-iii)are equivalent is identical to the proof for Theorem II.5.6 for pseudometric spaces. That proof was deliberately worded in terms of open sets, closed sets, and neighborhoods so that it would carry over to this new situation.

"
" iiiÑÊ iv) E©0 Ò0ÒEÓÓ©0 Ò cl] Ð0ÒEÓÑÓ . Since cl ] Ð0ÒEÓÑ is closed in ]ß iii) tells
"
" us that 0Ò cl] Ð0ÒEÓÑÓ is a closed set in \ that contains EÞ Therefore cl\ ÐEÑ©0Ò cl ] Ð0ÒEÓÑÓ , socl0Ò\ ÐEÑÓ©Ò cl ] Ð0ÒEÓÑÓÞ

ivÑÊ i) Suppose +−\ and that R is a neighborhood of 0Ð+ÑÞ Let Oœ\
0ÒRÓ
"
" and Yœ\
cl\ O©0ÒRÓÞY is open, and we claim that +−Y
which will show that 0ÒRÓ
" is a neighborhood of + , completing the proof. So we need to show that +ÂOcl\ . But this is clear because: if +−O cl\ , then using iv) would give us that 0Ð+Ñ−0Òcl\ OÓ© cl ] 0ÒOÓ , which is impossible since R∩0ÒOÓœgÞñ

135 Example 8.3 Sometimes we want to know whether a certain property is “preserved by continuous functions”
that is, if \ has property T and 0À\Ä] is continuous and onto , must the image ] also have the property T ? For example, condition iv) in Theorem 8.2 implies that continuous maps preserve separability . Suppose H is a countable dense set in \ . Then 0ÒHÓ is countable and 0ÒHÓ is dense in ] because ] œ 0Ò\Ó œ 0Ò cl HÓ © cl 0ÒHÓÞ By contrast, continuous maps do not preserve first countability : for example, let Ð]ßg Ñ be any topological space. Let gw be the discrete topology on ] . Ð]ß g w Ñ is first countable and the identity map 3ÀÐ]ßgw ÑÄÐ]ß g Ñ is continuous and onto. Thus, every space Ð]ßg Ñ is the continuous image of a first countable space. Do continuous maps preserve other properties that we have studied
such as Lindelo¨f, second countable, or metrizable?

The following theorem makes a few simple and useful observations about continuity.

Theorem 8.4 Suppose 0ÀÐ\ßg ÑÄÐ]ß g w Ñ .

w 1) Let Fœ ran Ð0Ñ©]Þ Then 0 is continuous iff 0ÀÐ\ßÑÄÐFßg g F Ñ is continuous. In other words, Fœ the range of f Ð a subspace of the codomain ]Ñ is what matters for thecontinuity of 0 ; points of ] not in F (if any) are irrelevant. For example, the function sin ÀÄ‘ ‘ is continuous iff the function sin ÀÄÒ
"ß"Ó ‘ iscontinuous.

2) Let E©\ . If 0 is continuous, then 0lEœ1ÀEÄ] is continuous Þ That is, the restriction of a continuous function to a subspace is continuous. For example, sinÀÄ‘ ‘ is continuous, so sin ÀÄÒ
"ß"Ó is continuous. (The second function is really sinl , but it's abbreviated here to just “sin”.

3) If f is a subbase for gw (in particular, if f is a base), then 0 is continuous iff 0ÒYÓ
" is open whenever Y − f . In other words, t o check continuity, it is sufficient to show that the inverse image of every =?, basic open set is open.

Proof 1) Exercise: the crucial observation is that if S©] , then 0
" ÒSÓœ0
" ÒF∩SÓ .

2) If S is open in ] , then 1ÒSÓœ0ÒSÓ∩E

"
" which is an open set in E .

3) Exercise: the proof depends only on the definition of a subbase and set theory:

"α
" α and 0 Ò Yα À −EÓœ Ö0 ÒYÓÀ α −E×
"α
" α 0 Ò Yα À −EÓœ Ö0 ÒYÓÀ α −E× ñ Example 8.5

1) For any topological spaces \] and , every constant function 0À\Ä] must be
" continuous. ( Suppose 0ÐBÑœC! for all B . If S is open in ] , then 0ÒSÓœ ? ) If \ has the discrete topology and ] is any topological space, then every function 0 À\ Ä] is continuous.

136 2) Suppose \ has the trivial topology and that 0À\Ä0‘ . If is not constant, then there are points +ß,−\ for which 0Ð+ÑÁ0Ð,ÑÞ Let M be an open set in ‘ containing 0Ð+Ñ but not 0Ð,ÑÞ Then 0ÒMÓ
" is not open in \ so 0 is not continuous. We conclude that 0 iscontinuousiff 0 is constant. In this example, we could replace ‘ by any metric space Ð] ß .Ñ ; or, for that matter, by any topological space Ð]ßg Ñ that has what property?

3) Let \ be a rectangle inscribed inside a circle ] centered at T+−\ . For , let 0Ð+Ñ be the point where the ray from T through + intersects ] . ( The function 0 is called a “central projection.” ). Then both 0À\Ä] and 0
" À]Ä\ are continuous bijections.

Example 8.6 (Weak topologies) Suppose \ is a set. Let Y œÖ0À−E×α α be a collection of functionswhere each 0À\ÄÞα ‘ If we put the discrete topology on \ , then all of the functions 0α will be continuous. But a topology on \ smaller than the discrete topology might also make all the 0α 's continuous. The smallest topology on \ that makes all the given 0 α 's continuous is called the weak topology g on \ generated by the collection Y.

How can we describe that topology more directly? g makes all the 0α 's continuous iff for each
" open S©‘ and each α −E , the set 0ÒSÓα is in g . Therefore the weak topology generated by Y is the smallest topology that contains all these sets. According to Example 5.17.5, this means
" that weak topology g is the one for which the collection ÆœÖ0α ÒSÓÀS open in ‘α , −E× is a subbase. ( It is clearly sufficient here to use only basic open setsS from ‘
that is, open intervals Ð+ß,ÑÀ why ? Would using all open sets S put any additional sets into g ? )

# For example, suppose \œ‘ and that Y11 œÖß×" # contains the two projection maps
" 1"ÐBßCÑœB and 1 # ÐBßCÑœC . For an open interval YœÐ+ß,Ñ©‘ , 1 " ÒYÓ is the “open
" vertical strip” Y‚‘ ; and 1# ÒZÓ is the “open horizontal strip” ‘ ‚Z . Therefore a subbase for the weak topology on ‘# generated by Y consists of all such open horizontal or vertical strips. Two such strips intersect in an “open box” Ð+ß,Ñ ‚ Ð-ß.Ñ in ‘# , so it is easy to see that the weak topology is the product topology on ‘‚ ‘ , that is, the usual topology of ‘ # .

Suppose E©‘ and that 3ÀEÄ ‘ is the identity function 3ÐBÑœBÞ What is the weak topology on the domain E generated by the collection Y œÖ3× ?

137 Definition 8.7 A mapping 0ÀÐ\ßg ÑÄÐ]ß g w Ñ is called

open if whenever S is open in \ , then 0ÒSÓ is open in ] , and closed if whenever J is closed in \ , then 0ÒJÓ is closed in ] .

Suppose l\l" . Let g be the discrete topology on \ and let g w be the trivial topology on \ . The identity map 3ÀÐ\ßg ÑÄÐ\ß g w Ñ is continuous but neither open nor closed, and 3ÀÐ\ßgw ÑÄÐ\ß g Ñ is both open and closed but not continuous. Open and closed maps are quite different from continuous maps
even whenthe mapping is a bijection! Here are some examples that are more interesting.

Example 8.8

1) 0ÀÒ!ß#ÑÄW1" œÖÐBßCÑ− ‘ ### ÀB1Cœ"× given by 0ÐќР))) cos ,sin Ñ . It is easy to check that 0 is continuous, one-to-one, and onto. The set JœÒ1 ß# 1 Ñ is closed in Ò!ß#Ñ111 but 0ÒÒß#ÑÓ is not closed in W" . Also, Ò!ßÑ 1 is open in Ò!ß#Ñ 11 but 0ÒÒ!ßÑÓÓ is not open in W " . A continuous, one-to-one, onto mapping does not need to be open or closed x

2) Suppose\] and are topological spaces and that 0À\Ä] is a bijection . Then there is an inverse function 1œ0À]Ä\
" , and 0
" is continuous iff 0 is open. To check this, consider an open set Ñ in \ . Then C−1
" ÒSÓ iff 1ÐCÑ−S iff Cœ0Ð1ÐCÑÑ−0ÒSÓ , so 0ÒSÓœ1ÒSÓ
". So 0ÒSÓ is open iff 1ÒSÓ
" is open. So, fora bijection 00 , is open iff 0
" is continuous. If S is replaced in the argument by a closed set J © \ , then similar reasoning shows that a bijection 0 is closed iff 0
" is continuous. In part 1), the bijection 0 is not open and therefore 0ÀWÄÒ!ß#Ñ
" " 1 is not continuous. (Explain directly, without part 2), why 0
" is not continuous. )

Definition 8.9 A mapping 0ÀÐ\ßÑÄÐ]ßÑg g w is called a homeomorphism if 0 is a bijection and 0 and 0
" are both continuous. If a homeomorphism 0 exists, we say that \ and ] are homeomorphic and write\ ¶ ] .

Note: The term is “home omorphism,” not “homomorphism” (a term from algebra)Þ The etymologies are closely related: “-morphism” comes from the Greek word Ð. 9 39( ´ Ñ for “shape” or “form.” The prefixes “homo” and “homeo” come from Greek words meaning same” and “similar” respectively. There was a major dispute in western religious history, mostly during the 4>2 century AD, that hinged on the distinction between “homeo” and “homo.”

As noted in the preceding example, we could also describe a homeomorphism as a “continuous open bijection” or a “continuous closed bijection.”

It is obvious that among topological spaces, homeomorphism is an equivalence relation, that is, for topological spaces \ß]ß and ^À

i) \ ¶ \ ii) if \¶] , then ] ¶\ iii) if \¶] and ]¶^ , then \¶^ .

138 Example 8.10

1) The function 0ÀÒ!ß#ÑÄW1" given by 0ÐќР))) cos ,sin Ñ is not a homeomorphism even though 0 is continuous, one-to-one, and onto.

2) The “central projection” from the rectangle to the circle (Example 8.5.3 ) is a homeomorphism.

3) It is easy to see that any two open intervals Ð+ß ,Ñ in‘ are homeomorphic (just use a linear map of one interval onto the other). 1 1 The mapping tanÀ Ð
# ß # Ñ Ä ‘ is a homeomorphism, so that each nonempty open interval in ‘ is actually homeomorphic to ‘ itself.

3) If 0 ÀÐ\ß.ÑÄÐ]ß=Ñ is an isometry (onto) between metric spaces, then both 0 and 0
" are continuous, so 0 is a homeomorphism.

w 4) If .. and are equivalent metrics (so g. œÑ g . w , then the identity map 3ÀÐ\ß.ÑÄÐ\ß.w Ñ is a homeomorphism. Notice, however, that 3 doesn't preserve distances (unless . œ . w ). In general, a homeomorphism between metric spaces need not be an isometry. But, of course, an isometry is automatically a homeomorphism

" " 5) The function 0ÀÖÀ8−×Ä8 given by 0ÐÑœ88 is a homeomorphism Ð both spaces have the discrete topology! )Topologically , these spaces are the identical: both are just countable infinite sets with the discrete topology. 0 is not an isometry In general, two discrete spaces \ and ] are homeomorphic iff they have the same cardinality:any bijection between them is a homeomorphism. Roughly speaking, “size is the only possible topological difference between two discrete spaces.”

139 6) Let T denote the “north pole”of the sphere

W# œÖÐBßCßDÑÀB ### 1C 1D œ"ש‘ $ Þ

The function 0 illustrated below is a “stereographic projection”. The arrow starts at T , runs for a while inside the sphere and then exits through the surface of the sphere at a point ÐBß Cß DÑ . Let 0ÐBß Cß DÑ be the point where the tip of the arrow hits the BC -plane ‘# . In this way 0 maps each point in W
ÖT×# to a point in ‘ # . The function 0 is a homeomorphism. ( See the figure below. Consider the images or inverse images of open sets.)

In general, what is the significance of a homeomorphism 0 À\ Ä] ?

i) 0 is a bijection so it sets up a perfect one-to-one correspondence between the points in \]BÇCœ0ÐBÑ and : . We can imagine that 0 just “renames” the points in \ . There is also a perfect one-to-one correspondence between the subsets \α and ] α of \ and ] : \Ç]œ0Ò\Óα α α . Because 0 is a bijection, each subset ]©]α corresponds in this way to one and only one subset \α © \ .

ii) 0 is a bijection, so 0 treats unions, intersections and complements “nicely”:

a)
"α
" α 0 Ò ]α À −EÓœ Ö0 Ò]ÓÀ α −E× b)
"α
" α , and 0 Ò ]α À −EÓœ Ö0 Ò]ÓÀ α −E× c) α α 0Ò \α À −EÓœ Ö0Ò\ÓÀ α −E× d) α α and 0Ò \α À −EÓœ Ö0Ò\ÓÀ α −E×ß e) 0Ò\
GÓœ0Ò\Ó
0ÒGÓœ]
0ÒGÓ

(Actually a),b), c) are true for any function; but d) and e) depend on 0 being a bijection. )

These properties say that this correspondence between subsets preserves unions: if each then . Similarly, preserves intersection and \Ç]ßαα \Ç αα ]œ0Ò\Óœ0Ò\Ó α α 0 complements.

140 iii) Finally if 0 and 0
" are continuous, then open (closed) sets in \ correspond to open (closed) sets in ] and vice-versa.

The total effect is that all the “topological structure” in \ is exactly “duplicated” in ] and vice versa: we can think of points, subsets, open sets and closed sets in ] are just “renamed” copies of their counterparts in \ . Moreover 0 preserves unions, intersections and complements, so 0 also preserves all properties of \ that can defined be using unions, intersections and complements of open sets. For example, we can check that if 0 is a homeomorphism and E©\, then 0Ò int\] EÓœ int 0ÒEÓ , that 0Ò cl \] EÓœ cl 0ÒEÓ , and that 0Ò Fr \] EÓœ Fr 0ÒEÓÞ That is, 0 takes interiors to interiors, closures to closures, and boundaries to boundaries.

Definition 8.11 A property P of topological spaces is called a topological property if, whenever a space \ has property T] and ¶ \ , then the space ] also has property T .

If \] and are homeomorphic, then the very definition of “topological property” says that \ and ] have the same topological properties. Conversely , if two topological spaces \ and ] have the same topological properties, then \ and ] must be homeomorphic. ( Why? Let T be the property “is homeomorphic to \T .” is a topological property because if ]T has (that is, if ]¶\ ) and ^¶]ß^ then also has T . Moreover, \ has this property T , because \¶\ . So if we assume that ] has the same topological properties as \] , then has the property T] , that is, is homeomorphic to \ . )

So we think of two homeomorphic spaces as “topologically identical”
they are homeomorphic iff they have exactly the same topological properties. We can show that two spaces are not homeomorphic by naming a topological property of one space that the other space doesn't possess.

Example 8.12 Let T be the property that “every continuous real-valued function achieves a maximum value.” Suppose a space \ has property T and that 2À\Ä] is a homeomorphism. We claim that ] also has property T .

Let 0 be any continuous real-valued function defined on ] . Then 0‰2À\Ä‘ is continuous À ‘ 1 ß Å 0 \Ò ] 2

By assumption, 1 œ 0 ‰ 2 achieves a maximum value at some point + − \ , and we claim that 0 must achieve a maximum value at the point , œ 2Ð+Ñ − ]Þ If not, then there is a point C−] where 0ÐCÑ0Ð,Ñ . Let Bœ2
" ÐCÑÞ Then

1ÐBÑ œ 0Ð2ÐBÑÑÑ œ 0Ð2Ð2
" ÐCÑÑÑÑ œ 0ÐCÑ  0Ð,Ñ œ 0Ð2Ð+ÑÑ œ 1Ð+Ñ ,

which contradicts the fact that1 achieves a maximum value at + .

Therefore T is a topological property.

141 For example, the closed interval Ò!ß"Ó has property T discussed in Example 8.12 ( this is a well- known fact from elementary analysis
which we will prove later ). But Ð!ß"Ñ and Ò!ß"Ñ do not have this property T (why? ). So we can conclude that Ò!ß"Ó is not homeomorphic to either Ð!ß"Ñ or Ò!ß "ÑÞ

Some other simple examples of topological properties are cardinality, first and second countability, Lindelo¨f, separability, and (pseudo)metrizability. In the case of metrizability, for example:

If Ð\ß.Ñ is a metric space and 0ÀÐ\ß.ÑÄÐ]ßg Ñ is a homeomorphism, then we can define a metric .w on ] as .Ð+ß,Ñœ.Ð0 w
" Ð+Ñß0
" Ð,ÑÑ for +ß,−] . You then need to check that g. w œ g (using the properties of a homeomorphism and the definition of .w ). This shows that Ð]ßÑg is metrizable. Be sure you can do this!

9. Sequences

In Chapter II we saw that sequences are a useful tool for working with pseudometric spaces. In fact, sequences are sufficient to describe the topology in a pseudometric space
because the convergent sequences in Ð\ß .Ñ determine the closure of a set.

We can easily define convergent sequences in any topological space Ð\ßg Ñ . But, as we will see, sequences need to be used with more care in spaces that are not pseudometrizable. Whether or not a sequence ÐBÑ8 converges to a particular point B is a “local” question
it depends on “the B8's approaching nearer and nearer to B ” and ß in the absence of a distance function, we use the neighborhoods of B to determine “nearness to B .” If the neighborhood systemaB is “too large” or “too complicated,” then it may be impossible for a sequence to “get arbitrarily close” to B . Soon we will see a specific example where such a difficulty actually occurs. But first, we look at some of the things that do work out just as nicely for topological spaces as they do in pseudometric spaces.

Definition 9.1 SupposeÐBÑ8 is a sequence in Ð\ßÑg . We say that ÐBÑ8 converges to B if, for every neighborhood [ of Bb5− , such that B−[8 when 8 5 . In this case we write ÐB8 Ñ Ä B . More informally, we can say that ÐBÑÄB8 if ÐBÑ 8 is eventually in every neighborhood [ of B .

Clearly, we can replace “every neighborhood [ of B ” in the definition with “every basic neighborhood FB of ” or “every open set S containing B .” Be sure you are convinced of this.

In a pseudometric space a sequence can converge to more than one point, but we proved that in a metric space limits of convergent sequences must be unique. A similar distinction holds in topological spaces: the important issue is whether we can “separate points by open sets.”

142 Definition 9.2 Ð\ßÑXg is a " -space if whenever BÁC−\ there exist open sets YZ and such that B−Y , CÂY and C−ZßBÂZ ( that is, each point is in an open set that does not contain the other point) . Ð\ßg Ñ is a X# -space (or Hausdorff space ) if whenever B Á C − \ there exist disjoint open sets Y and Z such that B−Y and C−Z .

It is easy to check that i) \ is a X" -space iff for every B−\ÖB× , is closed, that ii) every X# -space is a X " -space iii) every metric space Ð\ß.Ñ is a X # -space (Hausdorff space)

There is a hierarchy of ever stronger “separation axioms” called XßXßXßXß!"#$ and X % that a topological space might satisfy. Eventually we will look at all of them. Each condition is stronger than the preceding ones in the listÐ for example, XÊXÑÞ# " The letter “X ” is used here because in the early (German) literature, the word for “separation axioms” was “Trennungsaxiome.”

Theorem 9.3 In a Hausdorff space Ð\ßg Ñ , a sequence can converge to at most one point.

Proof Suppose BÁC−\ . Choose disjoint open sets Y and Z with B−Y and C−Z . If ÐBÑÄB8, then ÐBÑ 8 is eventually in YÐBÑ , so 8 is not eventually in Z . Therefore ÐBÑ8 does not also converge to C . ñ

When we try to generalize results from pseudometric spaces to topological spaces, we often get a better insight about where the heart of a proof lies. For example, to prove that limits of sequences are unique it is the Hausdorff property that is important, not the presence of a metric. Here is another example: for a pseudometric space Ð\ß.Ñ we proved that B−E cl iff B is a limit of a sequence Ð+8 Ñ in E . That proof (see Theorem II.5.18 ) used the fact that there was a countable neighborhood base ÖFÐBÑÀ8−×" at each point B . We can see now that the countable 8 neighborhood base was the crucial fact
because we can prove the same result in any first countable topological space Ð\ßg Ñ .

But first, two technical lemmas are helpful.

Lemma 9.4 Suppose ÖRßRßÞÞÞßRßÞÞÞ×" # 5 is a countable neighborhood base at +−\ . Define Yœ5int ÐR∩ÞÞÞ∩RÑ " 5 . Then ÖYßYßÞÞÞßYßÞÞÞ× "#5 is also a neighborhood base at + .

Proof R"∩ÞÞÞ∩R 5 is a neighborhood of + , so +− int ÐR∩ÞÞÞ∩RÑœYÞ" 55 Therefore Y 5 is a open neighborhood of +R . If is any neighborhood of ++−R©R , then 5 for some 5 , so then +−Y©R©R5 5. Therefore the Y 5 's are a neighborhood base at +Þñ

The exact formula for the Y5 's in Lemma 9.4 doesn't matter; the important thing is that we get a “much improved” neighborhood base ÖYßYßÞÞÞßYßÞÞÞ×
"#5 one in which the Y 5 's are open and YªYªÞÞÞªYªÞÞÞ" # 5 . This new neighborhood base at + plays a role analogous to the neighborhood base F" Ð+ѪF" Ð+ѪÞÞÞªF " Ð+ѪÞÞÞ in a pseudometric space. We call # 5 ÖY" ßY # ßÞÞÞßY 5 ßÞÞÞ× an open, shrinking neighborhood baseat +Þ

143 Lemma 9.5 Suppose ÖY" ßY # ßÞÞÞßY 5 ßÞÞÞ× is a shrinking neighborhood base at B and that +−Y8 8 for each 8 . Then Ð+ÑÄBÞ 8

Proof If [ is any neighborhood of B , then there is a 5 such that B−Y©[5 . Since the Y 5 's are a shrinking neighborhood base, we have that for any 8 5ß+8 −Y 8 ©Y 5 ©[ . So Ð+8 ÑÄBÞ ñ

Theorem 9.6 Suppose Ð\ßÑg is first countable and E© \. Then BE − cl iff there is a sequence Ð+Ñ8 in E such that Ð+ÑÄB 8 . ( More informally, “sequences are sufficient” to describe the topology in a first countable topological space. )

Proof (É ) Suppose Ð+Ñ8 is a sequence in E and that Ð+ÑÄB8 . For each neighborhood [ of BÐ+Ñ, 8 is eventually in [ . Therefore [∩EÁg , so B−EÞ cl ( This half of the proof works in any topological space: it does not depend on first countability. )

(Ê ) Suppose B−EÞ cl Using Lemma 9.4, choose a countable shrinking neighborhood base ÖYßÞÞÞßYßÞÞÞ×" 8 at BÞ Since B−E cl , we can choose a point +−Y∩E8 8 for each 8 . By Lemma 9.5, Ð+8 ÑÄB . ñ

We can use Theorem 9.6 to get an upper bound on the size of certain topological spaces, analogous to what we did for pseudometric spaces. This result is not very important, but it illustrates that in Theorem II.5.21 the properties that are really important are “first countability” and “Hausdorff,” not the actual presence of a metric . .

Corollary 9.7 If H is a dense subsetin a first countable Hausdorff space Ð\ßg Ñ , then l\l Ÿ lHli! . In particular, If \ is a separable, first countable Hausdorff space, then i! l\lŸi! œ-Þ

Proof \ is first countable ß so for each B−\ we can pick a sequence Ð.ÑH8 in such that

Ð.ÑÄB8 ; formally, this sequence is a function 0ÀÄH0−HÞB , so B Since \ is Hausdorff, a sequence cannot converge to two different points: so if BÁC−\ , then 0Á0B C . Therefore i! the function FÀ\ÄH given by F ÐBÑœ0B is one-to-one, so l\lŸlHlœlHlÞñ

The conclusion in Theorem 9.6 may not be true if \ is not first countable: sequences are not always “sufficient to describe the topology” of \
that is, convergent sequences cannot always determine the closure of a set.

Example 9.8 (the space P )

th Let PœÖÐ7ß8ÑÀ7ß8−ß7ß8 !×™ , and let G4 be “the 4 column of P ,” that is GœÖÐ4ß8Ñ−PÀ8œ!ß"ßÞÞÞ×Þ4 We put a topology on P by giving a neighborhood base at each point : À

ÖÐ7ß 8Ñ× if : œ Ð7ß8Ñ Á Ð!ß!Ñ U: œ {FÀÐ!ß!Ñ−F andC4
F is finite for all but finitely many 4× if :œÐ!ß!Ñ

144 (Check that this definition satisfies the conditions in the Neighborhood Base Theorem 5.2 and therefore does describe a topology for P . )

If :ÁÐ!ß!Ñ then : is isolated in P . A basic neighborhood of Ð!ß!Ñ is a set which contains Ð!ß!Ñ and which, we could say, contains “most of the points from most of the columns.” With this topology, P is a Hausdorff space.

Certainly Ð!ß!Ñ− cl ÐP
ÖÐ!ß!Ñ×Ñ , but no sequence from P
ÖÐ!ß!Ñ× converges to Ð!ß!ÑÞ To see this, consider any sequence Ð+8 ) in P
ÖÐ!ß!Ñ× :

i) if there is a column G4! that contains infinitely many of the terms +8 ß then

RœÐP
GÑ∪ÖÐ!ß!Ñ×4! is a neighborhood of Ð!ß!Ñ and Ð+Ñ8 is not eventually in R.

ii) if every column G4 contains only finitely many + 8 's, then RœP
Ö+À8−×8 is a neighborhood of Ð!ß!Ñ and Ð+ 8 ) is not eventually in R (in fact, the sequence is never in R ).

In P , sequences are not sufficient to describe the topology: convergent sequences can't show us that Ð!ß!Ñ− cl ÐP
ÖÐ!ß!Ñ×Ñ . According to Theorem 9.6, this means that the countable space P cannot be first countable
there is a countable neighborhood base at each point : Á Ð!ß!Ñ but not at Ð!ß !ÑÞ

The neighborhood system at Ð!ß!Ñ “measures nearness to” Ð!ß!Ñ but the ordering relationship ЪÑamong the basic neighborhoods at Ð!ß!Ñ is very complicated
much more complicated than the neat, simple nested chain of neighborhoods Y" ªY # ªÞÞÞªY 8 ªÞÞÞ that could form a base at B in a first countable space. Roughly, the complexity of the neighborhood system is the reason why the terms of a sequence can't get “arbitrarily close” to Ð!ß !ÑÞ

Since sequences do suffice to describe the topology in a first countable space, it is not surprising that we can use sequences to determine the continuity of a function defined on a first countable space \ .

Theorem 9.9 Suppose Ð\ßÑg is first countable and 0ÀÐ\ßÑÄÐ]ßÑ g g w . Then 0 is continuous at +−\ iff whenever ÐBÑÄ+8 , then Ð0ÐBÑÑÄ0Ð+ÑÞ 8

Proof (Ê0 ) If is continuous at +[ and is a neighborhood of 0Ð+Ñ , then 0Ò[Ó
" is a
" neighborhood of + . Therefore ÐBÑ8 is eventually in 0Ò[Ó , so 0ÐÐBÑÑ8 is eventually in [Þ (This half of the proof is valid for any topological space \ : continuous functions always “preserve convergent sequences.” )

(É ) Let ÖYßÞÞÞßYßÞÞÞ×" 8 be a shrinking neighborhood base at + . If 0 is not continuous at + , then there is a neighborhood [0Ð+Ñ of such that for every 80ÒYÓ©Î[ , 8 .
" For each 8 , choose a point B−Y
0Ò[Ó8 8 . Then (since the Y 8 's are shrinking) we have ÐBÑÄ+8 but Ð0ÐBÑÑ 8 fails to converge to 0Ð+Ñ because 0ÐBÑ8 is never in [ñ . (Compare this to the proof of Theorem II.5.22. )

145 10. Subsequences

Definition 10.1 Suppose0 À Ä \ is a sequence in \ and that 9 À Ä is strictly increasing. The composition 0‰ÀÄ\9 is called a subsequence of 0 . 0 Ò \ 9Å ß 0 ‰ 9

If we write 0Ð8ÑœB8 and 9 Ð5Ñœ8 5 , then Ð0‰ÑÐ5Ñœ0Ð8ÑœB 9 58 5. We write the sequence 0 informally as ÐBÑ8 and the subsequence 0‰9 as ( BÑ8 5 . Since 9 is increasing, we have that 85 Ä∞ as 5Ä∞Þ

For example, if 9Ð5Ñœ85 œ#5 , then 0‰ 9 is thesubsequence written informally as

ÐBÑœÐBÑ85 #5 , that is, the subsequence ÐBßBßBßÞÞÞßBßÞÞÞÑÞ# % ' #5 But if 9 Ð8Ñœ 1 for all 8 , then 0 ‰ 9 is not a subsequence: informally, ÐB""" ßB ßB ßÞÞÞß B " ßÞÞÞÑ is not a subsequence of ÐBßBßBßÞÞÞßBßÞÞÞÑÞ"#$ 8 Every sequence 0 is a subsequence of itself: just let 9 Ð8Ñœ8 .

Theorem 10.2 Suppose B−Ð\ßÑg . Then ÐBÑÄB8 iffevery subsequence ÐBÑÄBÞ8 5

Proof (É ) This is clear because ÐBÑ8 is a subsequence of itself.

(Ê ) Suppose ÐBÑÄB8 and that ÐBÑ 8 5 is a subsequence. If [ is any neighborhood of BB−[, then 8 for all 88 some ! . Since the 8 5 's are strictly increasing, 885 ! for all

5some 5Þ! Therefore ÐBÑ 85 is eventually in [ ,so ÐBÑÄBÞñ 8 5

Definition 10.3 Suppose B−Ð\ßÑÞg We say that B is a cluster point of the sequence ÐBÑ8 if for each neighborhood [B of and for each 5 − , there is an 8  5 for which B−[8 . More informally, we say that B is a cluster point of ÐBÑ8 if the sequence is frequently in every neighborhood [ of B .

Definition 10.4 Suppose B−Ð\ßÑg and E©\Þ We say that B is a limit point of E if R∩ÐE
ÖB×ÑÁgfor every neighborhood RB
of that is, every neighborhood of B contains points of arbitrarily close to B but different from B .

Example 10.5 .

1) Suppose \œÒ!ß"Ó∪Ö#× and EœÖ#×Þ Then [∩EÁg for every neighborhood [## of , but is not a limit point of E
because [œÖ#× is a neighborhood of #\ in and [∩ÐE
Ö#×ÑœgÞ Each B−Ò!ß"Ó is a limitpoint of Ò!ß"Ó and also a limit point of \ . Since Ö#× is open in \# , is also not a limit point of \ .

2) Every point < in ‘ is a limit point of . If E©E , then has no limit points (in or in ‘).

3) If ÐBÑÄBß8 then B is a cluster point of ÐBÑÞ8 More generally, if ÐBÑ 8 has a

subsequence ÐBÑ85 that converges to BB , then is a cluster point of ÐBÑÞ8 (Why? )

146 4) A sequence can have many cluster points. For example, if the sequence Ð;8 Ñ lists all the elements of , then every <− ‘ is a cluster point of Ð;ÑÞ8

8 5) In ‘ , the sequence ÐBÑœÐÐ
"ÑÑ8 has exactly two cluster points:
" and " . But the set ÖBÀ8−לÖ
"ß"×8 has no limit points in ‘ Þ The set of cluster points of a sequence is not always the same as the set of limit points of the set of terms in the sequence ! (Is one of these sets always a subset of the other? )

Theorem 10.6 Suppose + is a cluster point of ÐBÑ8 in a first countable space Ð\ßÑg . Then there

is a subsequence ÐB85 ÑÄ+ .

Proof Let ÖY" ßY # ßÞÞÞßY 8 ßÞÞÞ× be a countable shrinking neighborhood base at +Þ Since ÐBÑ8 is frequently in Y" , we can pick 8 "8"8 so that B−Y" . Since ÐBÑ is frequently in Y # , we can pick an 88# " so that B−Y©YÞ 8# # " Continue inductively: having chosen 8@ÞÞÞ@8" 5 so that

B−Y©©Y855 ... " , we can then choose 88 51"5 so that B−Y©YÞ 851" 51"5 Then ÐBÑ 8 5 is a subsequence of ÐBÑ8 and ÐBÑÄ+Þ 8 ñ

Example 10.7 (the space P , revisited)

Let P be the space in Example 9.8 and let ÐBÑ8 be a sequence which lists all the elements of P
ÖÐ!ß!Ñ×Þ

Every basic neighborhood FÐ!ß!Ñ of is infinite, so F must contain terms B 8 for arbitrarily large 8 . This means that ÐBÑ8 is frequently in FÐ!ß!Ñ , so is a cluster point of ÐBÑÞ8

But no subsequence of ÐBÑ8 can converge to Ð!ß!Ñ
because we showed in Example 9.8 that no sequence whatsoever from P
ÖÐ!ß!Ñ× can converge to Ð!ß!ÑÞ Therefore Theorem 10.6 may not be true if the space \ is not first countable .

Consider any sequence ÐBÑÄÐ!ß!Ñ8 in P . If there were infinitely many BÁÐ!ß!Ñ8 , then we could form the subsequence that contains those terms, and that subsequence would be a sequence in P
ÖÐ!ß!Ñ× that converges to Ð!ß!Ñ
which is impossible. Therefore we conclude that eventually B8 œ Ð!ß!Ñ .

Suppose now that 0ÀPÄ is any bijection, and let 5œ0Ð!ß!ÑÞ

ñ whenever a sequence ÐBÑÄÐ!ß!Ñ8 in P, then Ð0ÐBÑÑÄ0Ð!ß!Ñœ5 8 in ( because, by the preceding paragraph, 0ÐB8 Ñ œ 0Ð!ß!Ñ œ 5 eventually) ñ the topology on is discrete so Ö5× is a neighborhood of 0Ð!ß!Ñ , but 0ÒÖ5×ÓœÖÐ!ß!Ñ×
" is not a neighborhood of Ð!ß!Ñ . Therefore 0 is not continuous at Ð!ß !Ñ

Theorem 9.9 does not apply to P : if a space is not first countable, sequences may be inadequate to check whether a function 0 is continuous at a point .

147 Exercises

E17. Suppose 0ß1ÀÐ\ßÑÄÐ]ßÑg" g # are continuous functions, that H is dense in \ , and that 0lHœ1lHÞ Prove that if ] is Hausdorff, then 0œ1Þ ( This generalizes the result in Chapter 2, Theorem 5.12.)

E18. A function 0 À Ð\ßg Ñ Ä ‘ is called lower semicontinuous if

0
" Ò,∞ÓœÖBÀ0ÐBÑ,×(, ) is open for every ,− ‘ , and 0 is called upper semicontinuous if

0
" [(
∞, , )] œÖBÀ0ÐBÑ@,× is open for each ,− ‘ .

a) Show that 0 is continuous iff 0 is both upper and lower semicontinuous.

b) Give an example of a lower semicontinuous 0 À ‘Ä ‘ which is not continuous. Do the same for upper semicontinuous.

c) Suppose E © \Þ Prove that the characteristic function ;E is lower semicontinuous ifE\ is open in and upper semicontinuous if E\ is closed in .

E19. Suppose \ is an infinite set with the cofinite topology, and that ] has the property that every singleton ÖC× is a closed set. ( You might want to check: this is equivalent to saying that ] is a X" -space: see Definition 9.2 ). Prove that if 0À\ Ä ] is continuous and onto, then either 0 is constant or \ is homeomorphic to ] .

1) Note: the problem does not say that if 0 is not constant, then 0 is a homeomorphism. 2) Hint: Prove first that if 0 is not constant, then l\lœl]lÞ Thenexamine the topology of ] .

E20. Suppose \ is a countable set with the cofinite topology. State and prove a theorem that completely answers the question: “what sequences in Ð\ßg Ñ converge to what points?”

E21. Suppose that Ð\ßg Ñ and Ð]ß g w Ñ are topological spaces. Recall that the product topology on \ ‚ ] is the topology for which the collection of “open boxes”

UœÖY‚ZÀY− g ßZ− g w × is a base.

a) The “projection maps” 1"À\‚]Ä\ and 1 # À\‚]Ä] are defined by

1"ÐBßCÑœB and 1 # ÐBßCÑœCÞ

We showed in Example 5.11 that 11"# and are continuous. Prove that 11" and # are open maps. Give examples to show that 1" and 1 # might not be closed.

148 b) Suppose that Ð^ßÑg ww is a topological space and that 0À^Ä\‚]Þ Prove that 0 is continuous iff both compositions 1"‰0À^Ä\ and 1 # ‰0À^Ä] are continuous. ( Informally: a mapping into a product is continuous iff its composition with each projection is continuous. )

c) Prove that ÐÐBßCÑÑÄÐBßCÑ−\‚]8 8 iff ÐBÑÄB8 in \ and ÐCÑÄC 8 in ] . ÐFor this reason, the product topology is sometimes called the “topology of coordinatewise convergence.” Ñ

d) Prove that \‚] is homeomorphic to ]‚\Þ (Topological products are commutative. )

e) Prove that Ð\‚]Ñ‚^ is homeomorphic to \‚Ð]‚^Ñ (Topological products are associative. )

E22. Let Ð\ßÑg and ( ]ßÑ f be topological spaces. Suppose 0À\Ä] . Let

>Ð0ÑœÖÐBßCÑ−\‚]ÀCœ0ÐBÑ× = “ the graph of 0 . ”

Prove that the map 2À\Ä> Ð0Ñ defined by 2ÐBÑœÐBß0ÐBÑÑ is a homeomorphism if and only if 0 is continuous.

Note: if we think of 0 as a set of ordered pairs, the “graph of 0 ” is 0 . More informally, however, the problem states a function is continuous iff its graph is homeomorphic to its domain.

E23. In Ð\ßÑg , a family of sets Y œÖFÀ−×α αA is called locally finite if each point B−\ has a neighborhood R such that R∩FÁgα for only finitely many α 's. ( Part b) was also in Exercise E2. )

a) Suppose (\ß .Ñ is a metric space and that Y is a family of closed sets. Suppose there

is an %! such that .FF (" ,# ) % for all FF− " , # YY . Prove that is locally finite.

b) Prove that if Y is a locally finite family of sets in Ð\ß g Ñ , then cl(FÑœ cl( FÑ . Explain why this implies that if all the F 's are closed, α−Eα α −E α α then F is closed. ( This would apply, for example, to the sets in part a). See also α−E α Exercise E2. )

c) (The Pasting Lemmas: compare Exercise II.E24 ) Let Ð\ßg Ñ and Ð]ß g w Ñ be topological spaces. For each α −E , suppose F©\α , that 0ÀFÄ] α α is a continuous function, and that 0lÐFαα ∩FÑœ0lÐF " "α ∩FÑ " for all α ß " −E Then 0œ0 is a function and 0ÀFÄ]Þ ( Informally: each pair of functions α−E α α 0α and 0 " agree wherever their domains overlap; this allows use to define 0 by “pasting together” all the “function pieces” )

i) Show that if all the Fα 'sare open, then 0 is continuous.

149 ii) Show that if there are only finitely many Fα 's and they are all closed, then 0 is continuous.( Hint: use a characterization of continuity in terms of closed sets. )

iii) Give an example to show that 0 might not be continuous when there are infinitely many Fα 's all of which are closed.

iv) Show that if Y is a locally finite family of closed sets, then 0 is continuous. (Of course, iv) Ê ii) ).

Note: the most common use of the Pasting Lemma is when the index set E is finite Þ For example, suppose

" L" ÀÒ!ß"Ó‚Ò!ß# ÓÄÐ\ß.Ñ is continuous ß and " L# ÀÒ!ß"Ó‚Ò# ß"ÓÄÐ\ß.Ñ is continuous, and " " LÐ>ß"# ÑœLÐ>ß # # Ñ for all >−Ò!ß"Ó

# L" is defined on the lower closed half of the box Ò!ß "Ó ß L # is defined on the upper closed half, " and they agree on the “overlap”
that is, on the horizontal line segment Ò!ß"Ó‚Ö×Þ# Part b) Ðor part c) Ñ says that the two functions can be pieced together into a continuous function # LÀÒ!ß"Ó ÄÐ\ß.Ñß where LœL" ∪L # , that is

" LÐ>ßCÑ" if >Ÿ # LÐ>ß CÑ œ "  LÐ>ßCÑ# if > #

150 Chapter III Review

Explain why each statement is true, or provide a counterexample. If nothing else is mentioned, \ and ] are topological spaces with no other properties assumed.

1. For every possible topology g , the space ÐÖ!ß"ß#×ß g Ñ is pseudometrizable.

2. A convergent sequence in a first countable topological space has at most one limit.

3. A one point set ÖB× in a pseudometric space Ð\ß.Ñ is closed.

w 4. Suppose g and g are topologies on \ and that for every subset E\ of , clg ÐEÑœ cl g w ÐEÑ . Then gœ g w Þ

5. Suppose 0À\Ä] and E©\ . If 0lEÀEÄ] is continuous, then 0 is continuous at each point of E .

6. Suppose f is a subbase for the topology on \ and that H©\ . If Y∩HÁg for every Y−f, then H is dense in \ .

7. If H is dense in Ð\ßÑgg and ‡ is another topology on \© with gg ‡ , then H is dense in Ð\ßg ‡ Ñ .

8. If 0À\Ä] is both continuous and open, then 0 is also closed.

9. Every space is a continuous image of a first countable space Þ

10. Let \œÖ!ß"× with topology g œÖgß\ßÖ×× 1 . There are exactly 3 continuous functions 0ÀÐ\ßg ÑÄÐ\ß g ÑÞ

11. If EF and are subspaces of Ð\ßÑg and both EF and are discrete in the subspace topology, then E ∪ F is discrete in the subspace topology.

12. If E©Ð\ßÑg and \ is separable, then E is separable.

151 13. If g is the cofinite topology on \ . Every bijection 0ÀÐ\ßÑÄÐ\ßÑg g is a homeomorphism.

14. If X has the cofinite topology, then the closure of any open set in X is open.

15. A continuous bijection from ‘ to ‘ must be a homeomorphism.

16. For every cardinal 7 , there is a separable topological space Ð\ßg ) with ±\±œ7 .

17. If every family of disjoint open sets in Ð\ßÑg is countable, then Ð\ßÑ g is separable.

# # # 18. For +−‘% and ! , let GÐ+ÑœÖB−% ‘%g À.ÐBß+Ñœ × . Let be the topology on ‘ for # # which the collection ÖGÐ+ÑÀ% %‘ !ß+− × is a subbasis. Let h be the usual topology on ‘ . Then the function 0ÀБg# ß ÑÄÐ ‘h # ßÑ given by 0ÐBßCÑœ (sin B ,sin C ) is continuous.

19. If H©‘ and each point of H is isolated in HH , then cl must be countable.

20. Consider the separation property W À every minimal nonempty closed set J is a singleton. ÐJ is a minimal nonempty closed set means: if E is a nonempty closed set and E©J , then EœJÑÞ\X If is a " -space, then \ has property W .

21. A one-to-one, continuous, onto map0ÀÐ\ßg ÑÄÐ]ß g w Ñ must be a homeomorphism.

22. An uncountable closed set in ‘ must contain an interval of positive length.

23. A countable metric space has a base consisting of clopen sets.

24. Suppose D©\ andthat H is dense in Ð\ßÑgg . If ‡ is topology on \ with gg ‡ © , then D is dense in Ð\ßg ‡ Ñ .

25. If H ©‘ and H is discrete in the subspace topology, then H is countable.

26. The Sorgenfrey plane has a subspace homeomorphic to ‘ (with its usual topology).

152 27. In , let U 8 œÖF© À8−Fand F© { "ß#ßÞÞÞß8××Þ At each point 8 , the U 8 's satisfy the conditions in the Neighborhood Base Theorem and therefore describe a topology on .

28. At each point 8− , let U8 œÖFÀFªÖ8ß81"ß81#ßÞÞÞ××Þ At each point 8 , the U 8 's satisfy the conditions in the Neighborhood Base Theorem and therefore describe a topology on .

29. Suppose Ð\ßÑg has a base U with llœ-Þ U Then \ has a dense set H with lHlŸ-Þ

30. Suppose Ð\ßÑg has a base U with llœ-Þ U Then at each point B−\ , there is a neighborhood base with lUB lŸ- .

31. Suppose \ is an infinite set. Let g" be the cofinite topology on \ and let g # be the discrete topology on \Þ If a function 0ÀÄÐ\ßÑ‘g" is continuous, then 0ÀÄÐ\ßÑ ‘g # is also continuous.

32. Let g be the “right-ray topology” on ‘g , that is, œÖÐ+ß∞ÑÀ+− ‘‘ ×∪Ög, × . The space Б ß g Ñ is first countable.

153