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 looked at some of the most basic 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.

Early in Chapter II we observed 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. The result was that we could carry over the definition of continuity from calculus to pseudometric spaces. The distance function . also led us to the idea of an open set in a pseudometric space. From there we developed properties of closed sets, closures, interiors, frontiers, dense sets, continuity, and sequential convergence.

One important observation was that open (or closed) sets are all we need to work with many of these concepts; that is, we can often do what we need using the open sets without knowing what specific . that generated these open sets: the topology g. is what really matters. For example, clEÐ\ß.ÑßE. is defined in in terms of the closed sets so cl doesn't change if is replaced with a different but equivalent metric .w one that generates the same open sets. Changing . to an equivalent metric .w also doesn't affect interiors, continuity, or convergent sequences. In summary: for many purposes .. is logically unnecessary (although might be a handy tool) after .Þ has done its job in creating the topology g.

This suggests a way to generalize our work. For a particular set \ we can simply add 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 these “open sets” to “behave the way open sets should behave.” 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 for each then S−α gα −Eß+−E S−+ g iii) if S"8 ß ÞÞÞß S −gg ß then S " ∩ ÞÞÞ ∩ S 8 − Þ

A set S©\ is called open if S−gg . The pair Ð\ß Ñ is called a topological space .

103 We emphasize that in a topological space there is no distance function . : therefore concepts like “distance between two points” and “%g -ball” make no sense in Ð\ß ÑÞ There is no preconceived idea about what “open” means; to say “SS− is open” means nothing more or less than “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 set J©Ð\ßgg Ñ is called closed if X J is open, that is, if \J− .

Definition 2.3 A point + − Ð\ßgg Ñ is called isolated if Ö+ } is open, that is, if Ö+× − Þ

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 is closed for each then is closed J+−EßJααα−E iii) if are closed, then 8 is closed. J"8 ß ÞÞÞß J3œ3 J 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 is sometimes possible to find a pseudometric . on \ for which gg. œ that is, a . which generates exactly the same open sets as those already given in g . But this cannot always be done.

Definition 2.5 A topological spaceÐ\ßg Ñ is calledpseudometrizable if there exists a pseudometric .\ on such that gg. œÞ. If is a metric, then Ð\ßÑ g is called metrizable.

Examples 2.6

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

At the opposite extreme, suppose gcœÐ\Ñ . Then g is called the discrete topology on \\ÞÐ\ßÑ and it is the largest possible topology on g 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 .\œÞ (or any equivalent metric) on , then gg. So a discrete topological space is metrizable.

2) Suppose \ œ Ö!ß "× and let gg œ Ögß Ö"×ß \× . Ð\ß Ñ is a topological space called

104 Sierpinski space. In this case it is not possible to find a pseudometric .\ on for which gg. œ , so Sierpinski space is not pseudometrizable. To see this, consider any pseudometric .\ on .

If .Ð!ß "Ñ œ ! , then . is the trivial pseudometric on \ and Ögß \× œgg. Á .

If .Ð!ß "Ñ œ$ggg  ! , then the open ball F$ Ð!Ñ œ Ö!× −.. ß so Á Þ (In this caseg. is actually the discrete topology: . is just a rescaling of the discrete unit metric.)

Another possible topology on \ œ Ö!ß " } is gggww œ Ögß Ö!×ß \× , although Ð\ß Ñ and Ð\ß Ñ seem very much alike: both are two-point spaces, each with containing 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 of what this means later in this chapter.

3) For a set \ , let gg œÖS©\À Sœg or \S is finite ×Þ is a topology on \À

i) Clearly, g−ggand \− .

ii) Suppose for each If then . S−αααgα −EÞαα−E Sœgß −E S− g Otherwise there is at least one SÁg . Then \S is finite, so \ S αα!!α−E α œ Ð\S Ñ©\S. Therefore \ S is finite, so S −g . ααα−Eαα! −E α −E α iii) If and some , then 88 so . S"8 ß ÞÞÞß S −gg S 4 œ g3œ" S 3 œ g 3œ" S 3 − Otherwise each S33"8 is nonempty, so each \S is finite Þ Then Ð\SÑ∪ÞÞÞ∪Ð\S Ñ 88 is finite, so . œ\3œ" S33 3œ" S −g In Ð\ßg Ñ , a set J is closed iff J œ g or J is finite. Because the open sets are g and the complements 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? ) In \Ð\ßÑ is infinite, then no point in g is isolated.

Suppose \ÞYZ is an infinite set with the cofinite topologyg If and are nonempty open sets, then \Y and \Z must be finite so Ð\YÑ∪Ð\ZÑœ\ÐY∩ZÑ is finite. Since \Y∩ZÁgÐÑ is infinite, this means that in fact, Y∩Z must be infinite . Therefore every pair of nonempty open sets in Ð\ßg Ñ has nonempty intersection!

The preceding paragraph lets us see that an infinite cofinite space Ð\ßg Ñ is not pseudometrizable: i) if .\Á is the trivial pseudometric on , then certainly gg. , and

ii) if .\ is not the trivial pseudometric on , then there exist points +ß , − \ for which .Ð+ß ,Ñ œ$  ! . In that case, F$$ Ð+Ñ and F Ð,Ñ ## would be disjoint nonempty open sets in ggg..ßÁÞ so

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

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

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

Each concept we defined for pseudometric spaces can be carried over directly to topological spaces if the concept was defined in topological terms ß that is 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 } \Eœ E \ œÖSÀS S©E× cl the closure of in is closed and \EœE \ œÖJÀJ J ªE× Fr\\\Eœ the frontier (or boundary) of E in \ œ clE∩ 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.

Theorem 2.8 Suppose E©Ð\ß.Ñ . Then

106 1) a) int EE©E is the largest open subset of (that is, if S is open and O , then S©int E). b) EEœEEE is open iff int (since int © , the equality is equivalent to E© int EÑ . c) B− int E iff there is an open set SS such that B− ©E

2) a) cl EEJJE is the smallest closed set containing (that is, if is closed and ª , then Jª cl E ). b) EEœEE©Eß is closed iff cl (since cl the equality is equivalent to clE©EÞ ) c) B− cl E iff for every open set SS containing B , ∩EÁg

3) a) FrEEœE is c losed and Fr Fr (X  ). b)B− Fr E iff for every open set SS containing B , ∩EÁg and S∩Ð\EÑÁg 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) clFrEœE∪ E 3) intFrclEœE Eœ\ Ð\EÑ 4) FrclintEœ E E 5) \œ int E∪ Fr E∪ int ÐX Ñ E , 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∪ cl F is the union of two closed sets, so cl E∪ cl F 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− cl E but BÂEÞ If S is any open set containing B , then S∩EÁg (because B− cl E ) and S∩Ð\EÑÁg Ð because the intersection contains B ). Therefore B−Fr E , so B−AA ∪ Fr Þ Conversely, suppose B−E∪ Fr E . If B−E , then B− cl EÞ And if BÂE , then B−Fr Eœ cl E∩ cl Ð\EÑ , so B− cl EÞ Therefore clAA œ ∪ Fr AÞ

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

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

107 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 \E which might created from 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 , \  cl Eß cl Ð\  cl Ð\  EÑÑß and so on. An old theorem of Kuratowski (1922) says that for any set EÐ\ßÑ in any space g , the upper bound is 14. Moreover, this upper bound is “sharp”E© there is a set ‘ from which 14 sets can actually be obtained! Can you find such a set?

Definition 2.10 Suppose H©Ð\ßgg Ñ . H is called dense in \ if cl Hœ\Þ The space Ð\ß 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 'ß ÞÞÞ×Þ

int„ œg , because any nonempty open set S has finite complement and therefore is not a subset of „‘ . In fact, intFœg whenever F is infinite.

cl„‘œÐßÑ because the only closed set containing „‘ is . Therefore ‘g is separable. In fact, any infinite set F is dense.

Fr„„œÐÑ∩ÐÑœ∩œ cl cl ‘„‘‘‘

Ðß‘g Ñ is not pseudometrizable (why? )

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

int ™ œg cl™‘œÐßÑ , so ‘g is separable Fr ™‘œ Any two nonempty open sets intersect, so Ðß‘g Ñ is not metrizable. Is it pseudometrizable?

3. Subspaces

108 If EÐ\ßÑE is a subset of a space g , then there is a natural way to make into a topological space. We use the open sets in \EÞ to define open sets in

Definition 3.1 Suppose E©\ , where Ð\ßg Ñ is a topological space. The subspace topology on E is defined as ggggEEœÖE∩SÀS− }. ÐEß Ñ is called a subspace of Ð\ß ÑÞ (.Check that ggE is actually a topology on EE To say that is a subspace of Ð\ßÑ implies that EÞE is given the topology gE To emphasize that is a subspace , we sometimes write E©Ð\ßg Ñ rather than just E©\ÞÑ

If E©\ , we sometimes refer to S∩E as the “restriction of S to E ” or the “trace of S on E ”

Example 3.2 Consider ‘©8− , where ‘ has its usual topology. For each  , the interval Ð8  "ß 8  "Ñ is open in ‘ . Therefore Ð8  "ß 8  "Ñ ∩ œ Ö8× 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 just what we would get if we used 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 ‘ .

Suppose E©Ð\ß.Ñ . Then we can think of two ways to make E into a topological space.

i) .\ gives a topology gg.. on . Take the open sets in and intersect them with EEÐÑ . This gives us the subspace topology on , which we call g. E and ÐEß Ðgg. ÑE Ñ is a subspace of Ð\ß ÑÞ

2) Or, we could treat E. as a pseudometric space: just use to measure distances in E.À\‚\Ä . To be very precise, ‘ , so we make a “new” pseudometric ..œ.lE‚EÐEß.Ñww defined by . Then w is a pseudometric space and we can w use .EÞ to generate open sets in : the topology g.w (Usually, we would be less compulsive about the notation and just use . also to refer to the pseudometric n EÞ But for just a moment it will help to distinguish . and .w œ .lE ‚ E with different names.)

It turns out (fortunately) 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©Ð\ß.Ñ , where . is a pseudometric: then Ðgg.. ÑE œ w .

Proof Suppose Y−Ðgg%.. ÑE , so YœS∩E where S− . Let E−Y . There is an ! such .w ..w that F%%% Ð+Ñ©SÞ Since . œ.lE‚E , we get that F Ð+ÑœF Ð+Ñ∩E©S∩EœY , so Y−g. w .

. w Conversely, suppose w For each there is an such that Y −g%.+ Þ +−Y ! F%+ Ð+Ñ©Yß and Y œ F..w Ð+ÑÞ Let S œ F Ð+Ñß an open set in g Þ Since F ..w Ð+Ñ œ F Ð+Ñ ∩ E , +−Y%%++ +−Y . %% ++ we get S ∩ E œ Ð F.. Ð+ÑÑ ∩ E œ Ð F Ð+Ñ ∩ EÑ +−S%%++ +−S œFÐ+ÑœYÞY−ÐÑÞñ.w Therefore g +−Y %+ . E Exercise Verify that in any space Ð\ßg Ñ

109 i) If YÐ\ßÑE is open in g and is an open set in the subspace YßE\Þ then is open in ()“An open subset of an open set is open.”

ii) If JÐ\ßÑE is closed in g 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 If R©Ð\ßg Ñ and B− int R , then we say that N is a neighborhood of B Þ The collection aB œÖR©\ÀR 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, Rœ \Þ 2) If RR−B−R∩Rœ"#B and a , then int " int # (why? ) int ÐR∩RÑ "# . Therefore R"#∩R −a B. www 3) If R−aaBB and R©R, then B− int R© int R , so R − (that is, if RBRBww contains a neighborhood of , then is also a neighborhood of .)

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.

In a pseudometric space, we use the % -balls centered at BB to measure “nearness” to . For example, saying that “every %‘ -ball in centered at B contains an irrational number” 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 .” That we can say it all these different ways isn't surprising since, in Ð\ß .Ñß open sets and neighborhoods 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 BB ) as the tool we use to talk about “nearness to .”

For example, suppose \B−\ßB has the trivial topology g . For any the only neighborhood of is \C\À : therefore every in is in every neighborhood of B the neighborhoods of B are unable to “separate” BC and 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−\Þ If B−[ then (since [[ is open ), is a neighborhood of BRœÖB× . is the smallest neighborhood of B . So every point BR has a neighborhood that excludes all other points CCÁB : for every , we could say “CRBC is not within the neighborhood of .” This is analogous to saying “ is not within % of BCRBßB” (if we had a pseudometric). Because no point is “within of ” we call isolated. The neighborhoods of B tell us this.

110 (Of course, if we prefer, we could use “YB is an open set containing ” instead of “RB is a neighborhood of ” 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, using just the (open) neighborhoods FÐBÑ" in a pseudometric space Ð\ß.Ñ is enough to let us talk about continuous functions at B . 8

We use 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 UaBB©B is called a neighborhood base at 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 UUBB must be a neighborhood of B , but the collection may be much simpler than the whole neighborhood system RÞB The crucial thing is that every neighborhood RB of must contain “basic neighborhood” FB of .

Example 4.3 At a point B−Ð\ß.Ñ , let

i) UB œFÐBÑ the collection of all balls %

ii) U%B œFÐBÑ the collection of all balls % , where is a positive rational

iii) UB œFÐBÑ− the collection of all balls " for n 8

iv) UaBBœ (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) and iii) might be more convenient: each of those gives a countable family UB as a neighborhood base. (If \œ‘U, for example, with the usual metric . , then the collections B in i) and iv) are uncountable collections.) Of the four, iii) is probably the simplest choice for UB .

Suppose we want to check whether some property that involving neighborhoods of B is true. Often all we need to do is to check whether the property holds for the simpler neighborhoods in 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 SFÐBÑ contains a set " from UB . 8

Similarly, B− cl E 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 an “efficient” choice of the “simplest possible” neighborhood base UB at each point B is desirable.

111 Definition 4.4 We say that a space Ð\ßg Ñ satisfies the first axiom of countability (or, more simply, that Ð\ßg Ñ 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 gg is the discrete topology on \ß then Ð\ß Ñ is first countable. In fact, at each point BœÖÖB××Þ , we can choose a neighborhood base consisting of a single set: UB

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

To see this, suppose that we had a countable neighborhood base at BÀ UB"8œ ÖF ß ÞÞÞß F ß ÞÞÞ×Þ For any C Á Bß ÖC× is closed ß so \  ÖC× is a neighborhood of B5B−F©F©\ÖC× . Therefore for some , int55 , so ∞∞∞ int . But int so int . CÂ8œ" F888 B− 8œ" F 8œ" F œÖB×

Taking complements, we get ∞ int \ÖBל\8œ" F8 ∞ int . Since int is finite (why ?), this would mean that œ8œ" Ð\ F88 Ñ \ F \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.

For a given topology ga on \ , all the neighborhood systems B are completely determined by the topology gU , but B is not. As the preceding examples illustrate, there are usually many possible choices for UB . (Can you describe all the spaces Ð\ßgU Ñ for which 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 : aUBBœÖR©\À bF− 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 × œ ÖS À aB − S bF −UB B − F © S× , that is, SS is open iff contains a basic neighborhood of each of its points.

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 ggœ Ögß Ö!×ß \×Þ However, describing all the sets in explicitly is often not the easiest way to go.

In this section we look at three important, alternate ways to define a topology on a set. All of these will be used throughout the remainder of 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

Suppose that at each point B−Ð\ßgU Ñ we have picked a neighborhood base B . As mentioned above, the collections UgB contain implicitly all the information about the topology: a set S is in iff O contains a basic neighborhood of each of its points. This suggests that if we start with 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 UUBBÀ the sets in each 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 .

Theorem 5.1 Suppose Ð\ßgU Ñ is a topological space and that B is a neighborhood base at B for each B−\. Then

1) UUBBÁg and F− ÊB−F©\

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

3) if F−UB , then bM such that B−M©F and, aC C−I, b BCC −U such that − B C © I

4) OOBBO−ÍaB−b−gU B such that B−© .

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

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

3) Let MFœ−© int. Then BI B and because I is open, I is a neighborhood of each its points CC−C−F . Since UUCC is a neighborhood base at , there is a set F©MCC such that .

4) ÊB : If O is open, then O is a neighborhood of each of its points . 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 SSñ is a neighborhood of each of its points, so is open.

113 The features of a neighborhood base in Theorem 5.1 are the crucial ones about the behavior of a neighborhood base at B . The next theorem tells us that, with this information, we can put a topology on a set by giving a 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 assign a collection UB of subsets of \ in such a way that conditions 1) - 3) of Theorem 5.1 are true.Define gUœaB−bF−BÞ {OO©\ : B such that −F© O } Then gU is a topology on \BÐ\ßÑÞ and B is now a neighborhood base at in g

Note: In Theorem 5.2, we do not ask that the UB's satisfy condition 4) of Theorem 5.1 because the set \ has no topology and condition 4) would be meaningless. Here, Condition 4) is our motivation for how to define g using the FB 's.

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

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

Suppose OO−−EB−−EB−gα for all . If { : α }, then O for some ααα ! αgU−E. By definition of , there is a set F−© − such that BBO © { O : α −E }, so !Bαα!  {:O }.  α αg−E −

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

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

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

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

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

114 1) For each B−‘U , certainly BB Ág and B−ÒBß,Ñ for each set ÒBß,Ñ− U 2) If FœÒBß,Ñ""## and FœÒBß,Ñ are in U B , then (in this example) we can choose FœF∩FœÒBß,Ñ−$"# $U B , 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 FCC œÒCß-Ñ−U and C−ÒCß-Ñ©MÞ

According to Theorem 5.2, g‘œÖS© À aB−Sb ÒBß ,Ñ − UB such that B − ÒBß ,Ñ © S× is a topology on ‘ and UB 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 − gU : that is, the sets in B turn out to be open (not merely neighborhoods of BÞ ÐThis 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, in the Sorgenfrey line, there is a neighborhood base UB consisting of clopen sets at each point BÞ

We can write Ð+ß -Ñ œ∞ Ò+ " 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: gg. § . Á

‘ is dense in the Sorgenfrey line: if B− , then every basic neighborhood ÒBß,Ñ of B 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 BÖÒBßBÑÀ8−× the collection 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 # U‘ÐBßCÑ is a neighborhood base at ÐBß CÑÞ A set S © is open iff: for each ÐBß CÑ − 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 ‘g## is dense, so ÐßÑ 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 G%% Ð:Ñ in condition 3), let I œ F Ð: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−>U% with C !ß let : 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 >U . In this topology, the sets in : turn out to be open neighborhoods of :.

The space >> , with this topology, is called the “Moore plane .” Notice that is separable and 1st 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−gUUUgg is a union of sets from Þ More precisely, is a base if © and for each S− ß there exists a subfamily such that . We also call a base for the open sets and TU©Sœ T U we refer to the open sets in U as basic open sets .

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

Of course, we could choose Ugœ : every topology g is a base for itself. But there usually 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 gg is the only base for

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

Theorem 5.8 If Ð\ßgUg Ñ is a topological space with a base for , then

1) \œ ÖFÀF−U ×

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

Proof 1) Certainly But by definition of base, the open set is a union of a subfamily U ©\Þ \ of . Therefore UU œ\Þ

2) If FF−F∩F"# and U , then "# is open. If BFF−∩"# , the definition of base implies that there must be a set F−$ U such that B−©∩ F$"# F F . ñ

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 œÖOO©\À is a union of sets from UUלÖS©\À aB−SbF−such that B−F©S×. Then gUg is a topology on \ and is a base for .

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

If O ( ), then is a union of sets each of which is a union of sets αα−−EÖSÀ−E×gα α S α from . So clearly is a union of sets from . So . UαÖSαα À − E× Uαg ÖS À − E× −

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

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

Example 5.10 The collections

# U%‘%œÖFÐ: ÑÀ :−, !× w#" UœÖFÐ: 5 ÑÀ :−, 5− × and U‘ww œ ÖÐ+ß ,Ñ ‚ Ð-ß .Ñ À +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Ð In fact, all three are bases for the same topology on # that is, the usual topology 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\ßggwww Ñ and Ð] ß Ñ are topological spaces. Let U œ the set of “open boxes” in \‚] œÖY‚Z ÀY −ggwww and Z − ×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©EÞÐ Note that E itself might not be a “box.”Ñ

Let 11BBÀ\‚] Ä\ be the “projection” defined by ÐBßCÑœB . If Y is any open set in \ , " " then 1U1BBÒY Ó œ Y ‚ ] − . Therefore ÒY Ó is open. Similarly, for the other projection map " 11CCÀ\‚] Ä] defined by ÐBßCÑœC : if Z is open in ] , then 1C 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 the projection maps 11BC and are open maps.: that is, the images aof open sets in the product are open.)

117 If H\H]"# is dense in and is dense in , we claim that H‚H\‚] " is dense in . If ÐBß CÑ − E, where E is open, then there are nonempty open sets Y © \ and Z © ] for which ÐBßCÑ−Y‚Z©EÞ Since B−-6 D"" , Y∩H Ág ; and similarly Z∩H # ÁgÞ Therefore ÐY ‚ Z Ñ ∩ ÐH"# ‚ H Ñ œ ÐY ∩ H " Ñ ‚ ÐZ ∩ H # Ñ Á g, so E ∩ ÐH "# ‚ H Ñ Á gÞ Therefore ÐBß CÑ −cl ÐH"# ‚ H ÑÞ 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 .Ñ À +  ,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\Z] chosen from a base for ” 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. V are basic open sets from \ and from ]Þ

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

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

" Example 5.14 The collection U‘œÖÒBßB8 ÑÀ B− ß8− × is a base for the Sorgenfrey " topology on ‘ . But the collection ÖÒBß B 8 Ñ À B − ß 8 − × 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 Ug is a base for , then B− cl E iff each basic open set F containing B satisfies F∩EÁgÞ

w 2) If 0 À Ð\ß .Ñ Ä Ð] ß . Ñ and Ug is a base for the topology . 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\ßgÆ Ñ is a topological space. A family of open sets is called a subbase for the topology gU if the collection containing all finite intersections of sets from Æ is a base for g . (It is clear that if UgU is a base for , then is also a subbase for g .)

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

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

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

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

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

1) \−UÆ since \ is the intersection of the empty subcollection of (this follows the convention that the intersection of an empty family of subsets of \\ is itself. See Example I.4.5.5 ). Since certainly {: } \−UU ß \œ FF − Þ 2) Suppose FF−B−F∩F"# and U , and "# . We know that Fœ∩∩ ""7SS ... and F∩∩# œ S7" ... S 75 for some S " ,..., S 7 , ..., S 75 −Æ , so

B−F$"#" œF ∩F œ W ∩ ... ∩W 77"75 ∩W ∩ ... ∩W −U

Therefore gUUgœÖSÀS is a union of sets from × is a topology, and 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 ÆÆ . Therefore is a subbase for g . ñ

Example 5.17

1) Let „Æ„Æœ Ö#ß %ß 'ß ÞÞÞ × © and œ ÖÖ"×ß Ö#×ß ×Þ is a subbase for a topology on UÆ. A base for this topology is the collection of all finite intersections of sets in :

Uœ Ög , ß Ö#×ß Ö"×ß „ × . and the (not very interesting) topology gU generated the set of all possible unions of sets from :

gœ Ög , , Ö"}, {2}, {1,2}, „„, ∪ Ö"××

119 2) For each 8 −Æ ß let W88 œ Ö8ß 8  "ß 8  #ß ÞÞÞ×Þ The collection œ ÖW À 8 − × is a subbase for a topology on  . Here, W∩WœW87 5 where 5œ max Ö7ß8× , so the collection of all finite intersections from ÆÆ is just itself. So UÆœ is actually a base for a topology. The topology is gÆœ ∪ Ög× .

3) Let Æ‘‘‘œÖjÀj is a straight line in ## × in . If :− # , then Ö:× is the intersection of two sets from ÆUÆ , so Ö:× − Þ is a subbase for the discrete topology on ‘# .

4) Let Æ‘Æ‘œÖjÀj is a vertical line in ## ×Þ generates a topology on for which U‘œ Ö# ß g} ∪ Æ is a base and g œ ÖS À S is a union of vertical lines ×Þ

5) Suppose gÆggwww is any topology on \©Þ for which must contain the finite intersections of sets in Æ , and therefore must contain all possible unions of those intersections. Therefore ggÆw contains the topology for which 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, ww and w is a topology on . gggÆgœÖ À ª \×

Caution We said earlier that for some purposes it is sufficient to work with basic open sets, rather than arbitrary open setsB−E for example to check whether cl , or to check whether a function 0 between metric spaces is continuous. However, it is not always sufficient to work with subbasic open sets. Some caution is necessary. For example, Æ‘œÖMÀMœÐ∞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 " # ÂÞcl ™

D. The closure operator

Usually we describe a topology gÆUU by giving a subbase , a base , or by giving collections B to be the basic neighborhoods at each point B . In the early history of general topology, one other method was sometimes used. We will never 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 EEœEF is closed (by asking “is clg ? ”) and therefore can decide whether any set 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\ßg Ñ is a topological space and Eß F are subsets of \ . Then

1) clg gœg 2) E©clg E 3) EEœE is closed iff clg 4) clg Eœ clggÐE cl Ñ 5) clgggÐE∪ÑœB cl E ∪ cl F

Proof 1) Since ggœg is closed, clg

2) is closed and cl E© ÖJÀJ E©Jלg E 3) ÊE : If is closed, then E is one of the closed sets J used in the definition clggEœ ∩ÖJÀJ is closed and J ªE×ß so Eœ cl E. ÉEœE : If clgg, then E is closed because cl E is an intersection of closed sets.

4) clggggEEÐEÑ is closed, so by 3), clœ cl cl

5) EFEÐEªE∪ª , so clgg ∪FÑ cl. Likewise, cl gg ÐEª ∪FÑ cl B. Therefore clgggÐE∪Ñ F ª cl ÐE Ñ ∪ cl ÐF Ñ . On the other hand, clggE∪ cl F is a closed set that contains E∪ F , and therefore clggE∪ cl Fª cl g ÐE∪Ñ F . 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 \E©\ is a set and that for each , a subset clE is defined ( that is, we have a function cl À cc Ð\Ñ Ä Ð\Ñ ) in such a way that conditions 1), 2), 4) and 5) of Theorem 5.18 are satisfied. Define g œSÀ { clÐ\  SÑ œ \  S }. Then g is a topology on \ , and cl is now the closure operator for this topology (that is, clœ clg ).

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): since there is no topology on the given set \ , 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© cl E∪ cl F œ cl ÐE ∪FÑœ cl F g is a topology on the set \À Ð#Ñ i) \©\ cl , so \œ\Þ cl Therefore cl Ð\gÑœ\œ\œ\gg− cl , so g . (1) Since clÐ\\Ñœ cl gœgœ\\ , we have that \−g Þ

ii) Suppose Oα −−E−Egα for all . For each α! , we have

121 (*) clÐ\ OOOO Ñ © cl Ð\  Ñ œ \ because − g . This is true for every  αααα!!! , so clOOO ) . αA! −Ð\Ñ©Ð\œ\ααα  But by 2), we know thatOO cl . \αα © Ð\ Ñ Therefore clOOO , so . Ð\ ααα Ñ œ \  − g

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

With the topology g we now have a closure operator clgg . We want to show that clœ cl. First, observe that: (**) clg FœF iff F is closed in Ð\ßgg Ñ iff \F− iff cl Ð\Ð\FÑÑœ\Ð\FÑ iff clFœF .

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

E© clggg E , so (*) gives that cl E© cl Ð cl EÑ . But cl E is closed in Ð\ßg Ñ , so usingFœ clgggg E in (**) gives cl cl Eœ cl E . Therefore cl E© cl E . (4) On the other hand, cl clEEœ cl , so using FœE cl in (**) gives that clEÐ\ßÑÞ is closed in g (2) But clEªE , so cl E is one of the closed sets in the intersection that defines clggEEªEÞEœEñ . Therefore cl cl Therefore cl cl g .

Example 5.20 EEif is finite 1) Let \Eœ be a set. For E©\ , define cl .  \Eif is infinite Then cl satisfies the conditions in the Closure Operator Theorem. Since clEœE iff E is finite or Eœ\ , 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

clE œ ÖB −‘ À there is a sequence Ð+888 Ñ in E with each + B and | +  Bl Ä !× .

It is easy to check that cl satisfies the hypotheses of the Closure Operator Theorem. Moreover, a set EaB−Eb,BÒBß,Ñ©E is open in the corresponding topology iff such that . 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 \ .

Exercises

122 E1. Let \ œ Ö!ß "ß #ß Þß Þß Þ× œ Ö!× ∪9 . For S © \ , let S Ð8Ñ œ the number of elements in S∩Ò"ß8ÓlœlS∩[Define "ß8ÓlÞ g œÖSÀ !ÂS or Ð!−S and lim9SÐ8Ñ œ 1) × . 8Ä∞ 8

a) Prove that g is a topology on \ .

b) In any space Ð] ßg Ñ : a point B is called a limit point of the set E if R∩ÐEÖB×ÑÁg for every neighborhood R of B . Informally, this means that BE is a limit point of if there are points of EB other than itself arbitrarily close to B.) Prove that in any Ð] ßg Ñ , a subset F is closed iff F contains all of its limit points.

c) For Ð\ßg Ñ as defined above, prove that B is a limit point of \ if and only if B œ ! .

E2. Suppose thatE©Ð\ßÑα gα for each −E .

a) Suppose that cl is closed. Prove that  ÖEÀα α −E× ÖEÀ clαα −Eל cl ÖEÀ −E× αα

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

b) A family ÖFα Àαg − E× of subsets of Ð\ß Ñ is called locally finite if each point B−\ has a neighborhood R such that R∩Fα Ág for only finitely many αα 's. Prove that if ÖFα À − E× is locally finite, then

ÖFÀ clαα −Eל cl ÖFÀ −E× αα 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 UgÆ be a base for the topology = and let be a subbase " " for gU=Þ Prove or disprove: 0 is continuous iff 0ÒSÓ is open for all S− iff 0ÒSÓ is open for all S−Æ Þ

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

a) Give an example of a space Ð\ßg Ñ with a nontrivial topology and which is not a X" -space. b) Prove that \X is a " -space if and only if, given any two distinct points BßC−\ , each point is contained in an open set not containing the other point. c) Prove that in a XÖB×" -space, each set can be written as an intersection of open sets. d) Prove that a subspace of a XX"" -space is a -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 \]X and are " -spaces, so is \‚]Þ

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

123 E6. Suppose that Ð\ßgg Ñ and Ð] ßw Ñ are topological spaces. Recall that the product topology on \‚] is the topology for which the collection of “open boxes” UggœÖY‚Z ÀY − ßZ −w × is a base . Therefore a set S©\‚] is open in the product topology iff for all ÐBßCÑ−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 “product topology” is the topology on \‚] unless something different is explicitly stated. )

a) Verify that U is, in fact, a base for a topology on \‚] . b) Prove that the projection map 1B À\‚] Ä\ is an open map. 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 \] and each has a countable base iff \‚] has a countable base. Show that there is a countable neighborhood base at ÐBß CÑ − \ ‚ ] iff there is a countable neighborhood base UUBC at B−\ and a countable neighborhood base at C−]Þ e) Suppose Ð\ß ."# Ñ and Ð] ß . Ñ are pseudometric spaces. Define a pseudometric . on the set \‚] by .ÐÐB"" ß C Ñß ÐB ## ß C ÑÑ œ . ""# ÐB ß B Ñ  . #"# ÐC ß C ÑÞ Prove that the product topology on \‚] is the same as the topology g. . Note: . is the analogue of the taxicab metric in ‘#Þ Of course, there are other equivalent pseudometrics producing the product topology on \‚] , e.g., w # # "Î# . ÐÐB"" ß C Ñß ÐB ## ß C ÑÑ œ Ð. ""# ÐB ß B Ñ  . #"# ÐC ß C Ñ Ñ ß or ww . ÐÐB"" ß C Ñß ÐB ## ß C ÑÑ œ max Ö. ""# ÐB ß B Ñß . #"# ÐC ß C Ñ×Þ

E7. Ð\ßg Ñ is called a X#-space ( or Hausdorff space ) if whenever Bß C − \ and B Á C , then there exist disjoint open sets YZ and with B−YC−Z and .

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

124 E8. Suppose E©Ð\ßg ÑÞ The set E is called regular open if Eœ int Ð cl EÑ and E is regular closed if Eœ cl (int E ).

a) Show that for any subset F

i) \cl Fœ int Ð\FÑ ii) \int Fœ 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. Prove or give a counterexample:

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

b) In Ð\ßg Ñ , a finite set must be closed.

c) If for each , each is a topology on , then { : } is a topology αg−Eαα \ gα −E on \.

d) Ifgg"# and are topologies on XX , then there is a unique smallest topology g$ on such that ggg$"#ª∪ .

e) Suppose, for each αg−E , that α is a topology on \ . Then there is a unique smallest topology gαgg on \ªÞ such that for each , α

E10. Assume that each integer (except is divisible by a prime but nothing else about prime numbers. (This assumption is equivalent to assuming that each natural number bigger than can be factored into primes). For +− ™ and .− ß let

F+ß. œ ÖÞÞÞß +  #.ß +  .ß +ß +  .ß +  #.ß ÞÞÞ× œ Ö+  5. À 5 −™ × and let U™œÖF+ß. À+− ß.− × Ð=9U™ is the set of all arithmetic progressions in Ñ

a) Prove that Ug™ is a base for a topology on .

b) Show that each set is closed in (™g , )Þ

c) What is the set a prime number ? Explain why this set is not ÖF!ß: À : × closed in Ðß™g Ñ.

d) What does part c) tell you about the set of prime numbers?

125 6. Countability Properties of Spaces

Countable sets are generally 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 (or, is said to satisfy the first axiom of countability ) if we can choose a countable neighborhood base UB at every point B−\

second countable (or, is said to satisfy the second axiom of countability ) if there is a countable base Ug for the topology

separable if there is a countable dense set H\ in

Lindelo¨f if whenever is a collection of open sets for which , then there is hh œ\ a countable ww for which hhhœ ÖY"# ß Y ß ÞÞÞß Y 8 ß ÞÞÞ× © œ \Þ hhh is called an open cover of \\ , and w is called a subcove r from . Thus, is Lindelof¨ if “every open cover has a countable subcover .”)

Example 6.2

1) A countable discrete spaceÐ\ßgU Ñ is second countable because œ ÖÖB× À B − \× is a countable base.

2) ‘U is second countable because œÖÐ+ß,ÑÀ+ß,− × is a countable base. Similarly, 8 ‘U is second countable since the collection of boxes ÐB"" ß C Ñ ‚ ÐB ## ß C Ñ ‚ ÞÞÞ ‚ ÐB 88 ß C 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 gg . Ð\ß Ñ is second countable because we could choose Ugœ as a countable base.

The following theorem implies that each of the spaces in the preceding example is also first countable, separable, and Lindelof.¨ (However, it is really worthwhile to try to verify each of those assertions directly from the definitions.)

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

Proof Let Ugœ ÖS"# ß S ß ÞÞÞS 8 ß ÞÞÞ × be a countable base for .

i) For each 8B−SHœÖBÀ8−×H , pick a point 88 and let 8  . The countable set is dense. To see this, notice that if Y\8 is any nonempty open set in , then for some , B−S©Y88 so Y∩HÁg . So \ is separable.

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

iii) Let hh be any open cover of \B−\B− . If , then some set Y−ÞB For each B , we can then pick a basic open set S−BBBBUi such that B−S©Y . Let œÖSÀB−\×Þ Since each S−B Ui ß there can be only countably many different sets SB : that is, may contain

“repeats.” Eliminate repeats and list only the different sets in ii , so œ ÖSBB"# ß S ß ÞÞÞS B 8 ß ÞÞÞ × w where SBB88 © Y −hh Þ Every B is in one of the sets S B8"#8 , so œ ÖY BBB ß Y ß ÞÞÞY ß ÞÞÞ× is a countable subcover from h . Therefore \ñ is Lindelof.¨

The following examples show that no other implications exist among the countability properties mentioned 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\ßgh Ñ is Lindelof.¨ To see this ß let be an open cover of \ . Pick any one nonempty setY −h . Then \  Y is finite, say \  Y œ ÖB"8 ß ÞÞÞß B ×Þ For w each B3333 ß pick a set Y −hh with B − Y Þ Then œ ÖY × ∪ ÖY 3 À 3 œ "ß ÞÞÞß 8× is a countable (actually, finite ) subcover chosen from h .

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

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

g is a topology on \G©\ (check! ) called the cocountable topology. A set 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 Lindelof.¨ But Ð\ßg Ñ is not separable  every countable subset is closed and therefore not dense. By Theorem 6.3, Ð\ßg Ñ also cannot be second countable.

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

Ð\ßg Ñ is separable because Ö:× is dense.

Ð\ßgh Ñ is not Lindelof¨  because the cover œ ÖÖBß :× À B − \× has no countable subcover.

Is Ð\ßg Ñ first countable?

4) Suppose \\ is uncountable and let g be the discrete topology on .

127 Ð\ßgU Ñ is first countable because B œ ÖÖB×× is a neighborhood base at B

Ð\ßg Ñ is not second countable because each open set ÖB× would have to be in any base U .

Ð\ßg Ñ is not separable: for every countable set H , cl HœHÁ\Þ

Ð\ßgh Ñ is not Lindelof¨ because the cover œ ÖÖB× À B − \× has no countable subcover. (In fact, not a single set can be omitted from hh : has no proper subcovers at all. )

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. 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 Lindelof.¨

Proof i) Second countableÊ Lindelof:¨ by Theorem 6,3, this implication is true in any topological space.

ii) Lindelof¨¨ÊÐ\ß.Ñ8− separable: suppose is Lindelof. For each  , let h888œÖFÐBÑÀ" 8− ßB−\×. For each 8 , h is an open cover so h has a countable 8 " subcover8 that is, for each we can find countably many 8 -balls that cover \À say ∞ \œ F" ÐB8ß5 ÑÞ Let H be the set of centers of all these balls: HœÖB8ß5 À8ß5− ×Þ For 5œ" 8 " any B−\ and every 8 , we have B−F" ÐB8ß5 Ñ for some 5 , so .ÐBßB 8ß5 Ñ  so B can be 8 8 approximated arbitrarily closely by points from HH . Therefore is dense, so Ð\ß.Ñ is separable.

iii) SeparableÊÐ\ß.Ñ second countable: suppose is separable and that H œ ÖB"# ß B ß ÞÞÞß B 5 ß ÞÞÞ× is a countable dense set. Let UU œ ÖF" ÐB 5 Ñ À 8ß 5 − × . is a countable 8 collection of open balls and we claim Ug is a base for the topology .Þ

Suppose C−S−X. Þ By the definition of “open,” we have FÐCÑ©S% ""% for some % ! . Pick 8 so that 8#  ß and pick B55 −H so that .ÐBßCÑ 8 Þ Then C − F"" ÐB5555 Ñ © F% ÐCÑ © S (because D − F ÐB Ñ Ê .ÐDß CÑ Ÿ .ÐDß B Ñ  .ÐB ß CÑ 88 "" % #ÐÑœ88 # % ). 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 common to all pseudometric spaces (for example, first countability), or ii) show that it has one but not all of the properties “second countable,” “Lindelof,”¨ or “separable.”

128 Exercises

E11. Define g‘œÖY∪Z ÀY 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 Ðß‘g Ñ first countable? second countable? Lindelof¨ ? separable? The spaceÐß‘g Ñ is called the “scattered line.” We could change the definition of g by replacing ‘ with some other set E© ß creating a topology in which the set E is “scattered.” ∞ Hint: See Example I.7.9.6. It is possible to find open intervals MMª88 such that 8œ"  ∞  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\ÞG be the set of all condensation points in Prove that is closed. b) Prove that if \\G is second countable, then is countable.

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

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

a) Prove that a second countable space is hereditarily Lindelof.¨

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

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

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 cU and are two bases for the topology in Ð\ß g Ñ , and that cU and

129 are infinite.

a) Prove that there is a subfamily UUww©llŸll such that U is also a base and U w c . (Hint: For each pair >œÐYßZÑ−cc ‚ , 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 E©\ß where Ð\ßgg Ñ is a topological space. We defined the subspace topology E on EœÖE∩SÀS−× in Definition 3.1: ggE .

In this section we explore some simple but important properties of subspaces.

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

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

2) we can give FE the subspace topology gF , and then give the subspace topology from the space ÐFßggFF Ñ  that is, we can give E the topology Ð ÑE Þ

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 “a subspace of a subspace is a subspace.”

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

Proof Y−ggE iff YœS∩E for some S− iff YœS∩ÐF∩EÑ iff YœÐS∩FÑ∩E . But S∩F −ggFF , so the last equation holds iff S−Ð ÑE . ñ

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

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

Theorem 7.2 Suppose E©Ð\ßg Ñ . J is closed in E iff J œE∩G where G is closed in \ .

Proof J is closed in E iff EJ is open in E iff EJ œS∩E (for some open set S in \ ) iffJ œ E  ÐS ∩ EÑ œ Ð\  SÑ ∩ E œ G ∩ E (where G œ \  S is a closed set in \ ). ñ

Theorem 7.3 Suppose E©Ð\ßg Ñ .

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

2) If Ug is a base for , then ÖF ∩ E ÀF−U × is a base for gE Þ

Slightly abusing notation, we can informally write these collections as UU+ ∩E and. ∩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 + in E . Then +− intE R©R , so there is an open set S\ in such that +−RœS∩E©SÞ intE Since UU+ is a neighborhood base at +\ in , there is a neighborhood F−+ such that +−F©SÞ Then +−Ð int\+\ FÑ∩E©F∩E−U ∩E . Since Ð int FÑ∩E is open in E , we see that F∩E is a neighborhood of + in E . And since +−F∩E©S∩Eœ intE R ©R , we see that U+ ∩E is a neighborhood base at + in E . ñ

2) Exercise

Theorem 7.3 tells us that we can get a neighborhood base at a point +−E 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:−W©ÞœÖFÐ:ÑÀ!× is a circle in ‘‘U% and that : % is a #" neighborhood base at :∩W: in ‘U , and therefore : is a neighborhood base at in the subspace """ W∩WW:. The sets in U: are “open arcs on containing .” (See the figure,)

The following theorem relates closures in subspaces to closures in the larger space. It is a useful technical tool.

Theorem 7.6 Suppose EEœF∩EÞ©F©Ð\ßg Ñ, then clclF\

Proof F∩ cl\\F E is a closed set in F that contains Eß so F∩ cl Eª cl EÞ

On the other hand, suppose ,−F∩ cl\F E . To show that ,− cl E , pick an open set Y in F, that contains . We need to show Y∩EÁgS\ . There is an open set 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) œœ∩ cl‘ cl

2) clÐ!ß#Ó Ð!ß"Ñœ cl‘ Ð!ß"Ñ∩Ð!ß#ÓœÒ!ß"Ó∩Ð!ß#ÓœÐ!ß"Ó

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

œÁ∩œgint‘ int , and

gœFr‘ Á Fr ∩ œ Þ

Why does “cl” have a privileged role 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 TET , then every subspace also has property .

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 separable because # is dense. Consider the subspace H œ ÖÐBß CÑ À B  C œ "× . The set Y œ Ò+ß +  "Ñ ‚ Ò,ß ,  "Ñ is open in \ so if Ð+ß ,Ñ − Hß then Y ∩ H œ ÖÐ+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 Lindelof¨ property is not heredity. Let E: be an uncountable set and let be an additional point not in E\œE∪Ö:×Þ . Define Put a topology on \ by describing a neighborhood base at each point. U œ ÖÖ+×× for + − E +  U: œÖFÀ :−Fand \F is finite ×

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

If ii is an open cover of \:−ZZ− , then for some . 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−CCii with C−ZÞ Then œÖZ×∪ÖZÀC−\Z× C is a countable (in fact, finite) subcover from i , so \ is Lindelof.¨

The definition of U+ implies that each point of EEE is isolated in ; that is, is an uncountable discrete subspace and h œ ÖÖ+× À + − E× is an open cover of E that has no countable subcover. Therefore the subspace E is not Lindelof.¨

133 Even if a property is not hereditary, it is sometimes “inherited” by certain subspaces as the next theorem illustrates.

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

Proof Suppose O\œÖYÀ−E× is a closed subspace of the Lindelof¨ space . Let hαα be a cover of OY by sets α that are open in O . For each α , there is an open set Z\α in such that Z∩OœYÞαα

Since O is closed, \  O is open and iα œ ÖÖ\  O×× ∪ ÖZα À − E× is an open cover of \Þ w But \ is Lindelof,¨ so iii has a countable subcover from , say œ Ö\  Oß Z"8 ß ÞÞÞß Z ß ÞÞÞ×Þ (The set \O may not be needed in i w , but it can't hurt to include it. ) Clearly then, the collection ÖY"8 ß ÞÞÞß Y ß ÞÞÞ× is a countable subcover of O from h . ñ

(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 subcoverÞÑ

134 8. Continuity

To define continuous functions between pseudometric spaces, we began by using the distance function . . But then we proved that our definition is equivalent to other formulations stated in terms of open sets, closed sets, or neighborhoods. In the context of topological spaces we do not have distance functions to describe “nearness.” But we can still use neighborhoods of + to take about “nearness” to + .

Definition 8.1 A function 0ÀÐ\ßgg ÑÄÐ]ßw Ñ is continuous at + −\ if whenever R is a neighborhood of 0Ð+Ñ , then 0" ÒRÓ 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 Z0Ð+ÑY+ containing there is an open set containing such that 0ÒYÓ © Z

iii) for each basic open set Z0Ð+ÑY containing there is a basic open set containing +0ÒYÓ©Z such that .

The conditions i) - iii) for continuity in the following theorem 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\ßgg ÑÄÐ]ßw Ñ . The following are equivalent.

i) 0 is continuous ii) if S−ggw" ß then 0 ÒSÓ− (the inverse image of an open set is open ) iii) if J]0ÒJÓ\ is closed in , then " is closed in (the inverse image of a closed set is closed) 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.

" " iiiÑ Ê iv) 0 Ò cl]] Ð0ÒEÓÑÓ ª 0 Ò0ÒEÓÓ ª E . Since cl Ð0ÒEÓÑ is closed in ] ß iii) gives " " that 0 Ò cl] Ð0ÒEÓÑÓ is a closed set in \ that contains EÞ Therefore 0 Ò cl]\ Ð0ÒEÓÑÓ ª cl ÐEÑ , so Òcl]\ Ð0ÒEÓÑÓÓ ª 0Ò cl ÐEÑÓÞ

ivÑÊ i) Suppose +−\ and that R is a neighborhood of 0Ð+ÑÞ Let Oœ\0" ÒRÓ " and Yœ\ cl\ O©0 ÒRÓÞY 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 + cl\\ O . But this is clear: if +− cl O , we would have 0Ð+Ñ − 0Ò cl\] O Ó © cl 0ÒOÓ , which is impossible since R ∩0ÒOÓ œ gÞ ñ

135 Example 8.3 Sometimes we want to know whether a certain property is “preserved by continuous functions”\ that is, if has property T0À\Ä] and is continuous and onto , must ]T also have the property ? 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 0Òcl HÓ © cl 0ÒHÓÞ By contrast, continuous maps do not preserve first countability : for example, let Ð] ßg Ñ be any topological space. Let ggww be the discrete topology on ]Ð]ßÑ . is first countable and the identity map 3ÀÐ]ßggw ÑÄÐ]ß Ñ is continuous and onto. Thus, every space Ð]ß g Ñ is the continuous image of a first countable space. Do continuous maps preserve other properties such as Lindelof,¨ second countable, or metrizable?

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

Theorem 8.4 Suppose 0ÀÐ\ßgg ÑÄÐ]ßw Ñ.

w 1) Let Fœ ran Ð0Ñ©]Þ Then 0 is continuous iff 0ÀÐ\ßgg ÑÄÐFßF Ñ is continuous. In other words, Fœ the range of f Ð a subspace of the codomain ]Ñ is what matters for the continuity of 0Þ Points (if any) in ] F are irrelevant. For example, the function sinÀÄ‘‘ is continuous iff the function sin ÀÄÒ"ß"Ó ‘ is continuous.

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, sinÀ Ä Ò  "ß "Ó is continuous.)

3) If fg is a subbase for w" (in particular, if f is a base), then 00ÒYÓ is continuous iff is open whenever Y−f . To 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]1ÒSÓœ0ÒSÓ∩E is open in , then " " which is an open set in E . 3) Exercise: it depends only on the definition of a subbase and set theory: " " and 0 Ò Yαα Àαα −EÓœ Ö0 ÒYÓÀ −E× " " 0 Ò Yαα Àαα −EÓœ Ö0 ÒYÓÀ −E× ñ Example 8.5

1) If \] has the discrete topology and is any topological space, then every function 0À\Ä]is continuous.

2) Suppose \0À\Ä00 has the trivial topology and that ‘ . If is constant, then is continuous. If 0 is not constant, then there are points +ß , − \ for which 0Ð+Ñ Á 0Ð,ÑÞ Let M be an open set in ‘ containing 0Ð+Ñ but not 0Ð,ÑÞ Then 0" ÒMÓ is not open in \ so 000 is not continuous. We conclude that is continuous iff is constant. (In this example, we could replace ‘ by any metric space Ð] ß .Ñß or by any topological space Ð] ßg Ñ that has what property?

3) Let \]T+−\ be a rectangle inscribed inside a circle centered at . For , let

136 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 \œÖ0À−E× is a set. Let Yαα be a collection of functions where each 0À\ÄÞα ‘ If we put the discrete topology on \ , then all of the function 0\α will be continuous. But a topology on smaller than the discrete topology might also make all the 0\0α 's continuous. The smallest topology on that makes all the α 's continiuous 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©‘α and each −E , the set 0α ÒSÓ 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 gÆ‘α is the one for which the collection œÖ0α ÒSÓÀ S open in , −E× is a subbase. (It is clearly sufficient here to use only basic open setsS from ‘ that is, open intervals Ð+ß ,Ñ À why ? Would using all open sets S put any additional sets into g ? )

# For example, suppose \œ‘Y11 and that œÖBC ß × contains the two projection maps " 11BCÐBß CÑ œ B and ÐBß CÑ œ C . For an open interval Y œ Ð+ß ,Ñ © ‘1 , B ÒY Ó is the “open " vertical strip” Y‚‘1 ; and C ÒZÓ is the “open horizontal strip” ‘ ‚Z . Therefore a subbase for the weak topology on ‘Y# generated by 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 usuall topology of ‘# .

Suppose E©‘‘ and that 3ÀEÄ is the identity function 3ÐBÑœBÞ What is the weak topology on the domain EœÖ3× generated by the collection Y ?

137 Definition 8.7 0ÀÐ\ßgg Ñ Ä Ð]ßw Ñ is called open if whenever S is open in \ , then 0ÒSÓ is open in ]0 , and is called closed if 0ÒJÓ is closed in ] whenever J is closed in \ .

Suppose l\l  " . Let gg be the discrete topology on \ and let w be the trivial topology on \ . The identity map 3ÀÐ\ßgg ÑÄÐ\ßw Ñ is continuous but neither open nor closed, and 3ÀÐ\ßggw ÑÄÐ\ß Ñ is both open and closed but not continuous. Open (closed) maps are quite different from continuous maps0 even if is a bijection! Here are some examples that are more interesting.

Example 8.8

1) 0 À Ò!ß #1‘ Ñ Ä W"### œ ÖÐBß CÑ − À B  C œ "× given by 0Ð ))) Ñ œ Ð cos ,sin Ñ . It is easy to check that 0JœÒß#ÑÒ!ß#Ñ is continuous, one-to-one, and onto. 11 is closed in 1 but 0Ò Ò11 ß # ÑÓ is not closed in W" . Also, Ò!ß 1 Ñ is open in Ò!ß # 1 Ñ but 0Ò Ò!ß 1 Ñ Ó Ó 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 so there is an inverse function 0À]Ä\Þ" Then 0 " is continuous iff 0 is open. To check this, let 1 œ 0" Þ For an open set S © \ß we have C − 1 " ÒSÓ iff 1ÐCÑ − S iff C œ 0Ð1ÐCÑÑ − 0ÒSÓ , so 0ÒSÓ œ 1" ÒSÓ. So 0ÒSÓ is open iff 1 " ÒSÓ is open. For a bijection 0 , “ 0 is open” is equivalent to “0 " is continuous.” If SJ©\ is replaced by a closed set , then a similar argument also shows that a bijection 00 is closed iff " is continuous .

Definition 8.9 A mapping 0ÀÐ\ßgg ÑÄÐ]ßw Ñ is called a homeomorphism if 0 is a bijection and 00 and " are both continuous. If a homeomorphism 0 exists, we say that \] and are homeomorphic and write\¶] .

Note: The term is “homeomorphism," not “homomorphism” (a term from algebra)Þ The etymologies are closely related: “-morphism” comes from the Greek word Ð9.39( 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 in a collection of topological spaces, the homeomorphism relation “¶ ” is an equivalence relation: if \ß ] ß and ^ are topological spaces, then

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

138 Example 8.10

1) The function 0ÀÒ!ß#ÑÄW1)))" given by 0ÐќРcos ,sin Ñ is not a homeomorphism even though 0 is continuous, 1 1, and onto.

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

3) Using a linear map, it is easy to see that any two open intervals Ð+ß ,Ñ in‘ are 11 homeomorphic. The mapping tanÀÐ## ß ÑÄ‘ is a homeomorphism, so that each nonempty open interval in ‘‘ is actually homeomorphic to itself.

3) If 0 À Ð\ß .Ñ Ä Ð] ß =Ñ is an isometry (onto) between metric spaces, then both 0 and 00" are continuous, so is a homeomorphism.

w 4) If .. and are equivalent metrics (so gg.. œÑw , then the identity map 3 À Ð\ß .Ñ Ä Ð\ß .w Ñ is a homeomorphism. A homeomorphism 0 between metric spaces need not preserve distances, that is, 0 need not be an isometry.

"" 5) The function 0ÀÖ88 À8− ×Ä given by 0ÐÑœ8 is a homeomorphism Ðboth 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 Any two discrete spaces \] and with the same cardinality are homeomorphic: any bijection between them is a homeomorphism.

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

W# œÖÐBßCßDÑÀB ### C D œ"ש‘ $ Þ

The function 0T illustrated below is a “stereographic projection”. The arrow starts at , runs inside the sphere and exits through the surface of the sphere at a point ÐBßCßDÑ . 0ÐBßCßDÑ is the point where the tip of the arrow hits the BC -plane ‘## . Thus 0 maps each point in W  ÖT × to a point in ‘# . The function 0 is a homeomorphism. (See the figure below. )

What is the significance of a homeomorphism 0À\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 \]\]αα and of and : \Ç]œ0Ò\Óαα α. Because 0 is a bijection, each subset ]©] α corresponds in this way to one and only one subset \©\α .

ii) 00 is a bijection, so treats unions, intersections and complements “nicely”:

a) " " 0 Ò ]αα Àαα −EÓœ Ö0 Ò]ÓÀ −E× b) " " , and 0 Ò ]αα Àαα −EÓœ Ö0 Ò]ÓÀ −E× c) 0Ò \αα Àαα −EÓœ Ö0Ò\ÓÀ −E× d) and 0Ò \αα Àαα −EÓœ Ö0Ò\ÓÀ −E×ß e) 0Ò\GÓ œ 0Ò\Ó0ÒGÓ œ ] 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 α ]œ α 0Ò\Óœ0Ò\Ó α  α 0 complements.

140 iii) Finally if 00 and " are continuous, then open (closed) sets in \ correspond to open(closed) sets in ] and vice-versa.

The total effect is 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 \0 . Moreover 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 0E©\ is a homeomorphism and , then 0Òint\ EÓ œ int ] 0ÒEÓ , that 0Ò cl \] EÓ œ cl 0ÒEÓ , and that 0Ò Fr \] EÓ œ 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 \T]] has property and ¶\ , then the space also has property T .

By definition: if \] and are homeomorphic, then \] and have the same topological properties. Conversely , if two topological spaces \] and have the same topological properties, then \] and must be homeomorphic. (Why? Let Tœ “is homeomorphic to \ .” T is a topological property because if ]T has (that is, ]¶\^¶]ß^ ) and then also has T . And \T\¶\\] has this property , because . So if we are assuming that and have the same topological properties, then ]T]\ has the property , 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 showing 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 \T2À\Ä] property and that is a homeomorphism. We claim that ]T also has property .

Let 0] be any continuous real-valued function defined on . Then 0‰2À\Ä‘ is continuous À ‘ 1 ß Å0 \]Ò 2

By assumption, 1œ0‰2 achieves a maximum value at some point +−\ , and we claim that 0,œ2Ð+Ñ−]Þ must achieve a maximum value at the point If not, then there is a point C−] where 0ÐCÑ0Ð,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 simple examples of topological properties are: cardinality, first and second countability, Lindelof,¨ separability, and (pseudo)metrizability are topological properties. In the case of metrizability, for example:

If Ð\ß .Ñ is a metric space and 0 À Ð\ß .Ñ Ä Ð] ßg Ñ is a homeomorphism, then we can define a metric .w on ] as . w Ð+ß ,Ñ œ .Ð0 " Ð+Ñß 0 " Ð,ÑÑ for +ß , − ] . You then need to check that gg. w œ (using the properties of a homeomorphism and the definition of .Ð]ßÑw ). This shows that g 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 Ð\ß .Ñ  because convergent sequences determine the closure of a set.

We can easily define convergent sequences in any topological space Ð\ßg Ñ . But as we shall see, sequences need to be used with more care in spaces that are not pseudometrizable. Whether or not a sequence converges to a certain point B is a “local” question it depends on “getting near BB” and we “measure nearness” to by using the neighborhood system aaBB . If is “too large” or “too complicated,” then it may be impossible for a sequence to “get arbitrarily close” to B . We will see a specific example soon where such a problem 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 SupposeÐB88 Ñ is a sequence in Ð\ßg Ñ . We say that ÐB Ñ converges to B if, for every neighborhood [Bb5− of ,  such that B−[8 when nk . In this case we write ÐB888 Ñ Ä B. More informally, we can say that ÐB Ñ Ä B if ÐB Ñ is eventually in every neighborhood [B of .

Clearly, we can replace “every neighborhood [B of ” 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 situation holds in any space: the important issue is whether we can “separate points by open sets.”

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

It is easy to check that \X is a "-space iff for every B−\ÖB×Þ , is closed

There are several “separation axioms” called XßXßXßXß!"#$ and X % that a topological space might satisfy. We will look at all of them eventually. Each condition is stronger than the earlier ones in the listÐXÊXÑÞ for example, #" 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 ÐBÑÄB88, then ÐBÑ is eventually in Y , so ÐBÑ 8 is not eventually in Z . Therefore ÐBÑ 8 does not also converge to Cñ .

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. For a pseudometric space Ð\ß .Ñ we proved that B − cl E 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 ÖF" ÐBÑ À 8 8− × at each point B . We can see now that the countable neighborhood base was the crucial fact: we can prove the same result in any first countable topological space Ð\ßg Ñ .

First, two technical lemmas are helpful.

Lemma 9.4 Suppose ÖZ"# ß Z ß ÞÞÞß Z 5 ß ÞÞÞ× is a countable neighborhood base at + − \ and define Y5"5 œint ÐZ ∩ ÞÞÞ ∩ Z Ñ . Then ÖY "#5 ß Y ß ÞÞÞß Y ß ÞÞÞ × is also a neighborhood base at + .

Proof Z"5 ∩ ÞÞÞ ∩ Z is a neighborhood of + , so + − int ÐZ "55 ∩ ÞÞÞ ∩ Z Ñ œ Y Þ Therefore Y 5 is a open neighborhood of +R . If is any neighborhood of + , then +−Z©R5 for some 5 , so then +−Y55 ©Z ©R. Therefore the Y 5 's are a neighborhood base at +Þñ

The important thing is that we can get an “improved” neighborhood base ÖY"# ß Y ß ÞÞÞß Y 5 ß ÞÞÞ × in which the Y5"#5 's are open and Y ªY ªÞÞÞªY ªÞÞÞ ; the exact formula for the Y 5 's in Lemma 9.4 doesn't matter. This new neighborhood base at + plays a role like the neighborhood base F" Ð+Ñ ª F"" Ð+Ñ ª ÞÞÞ ª F Ð+Ñ ª ÞÞÞ in a pseudometric space. We call #5 ÖY"# ß Y ß ÞÞÞß Y 5 ß ÞÞÞ× an open, shrinking neighborhood base at +Þ

Lemma 9.5 Suppose ÖY"# ß Y ß ÞÞÞß Y 5 ß ÞÞÞ× is a shrinking neighborhood base at B and that + 8 − Y 8 for each 8Ð+ÑÄBÞ . Then 8

143 Proof If [B5B−Y©[Y is any neighborhood of , then there is a such that 55 . Since the 's are a shrinking neighborhood base, we have that for any 8 5ß+88 −Y ©Y 5 ©[ . So Ð+8 Ñ Ä BÞ ñ

Theorem 9.6 Suppose Ð\ßg Ñ is first countable and E©\. Then B − cl E iff there is a sequence Ð+88 Ñ in E such that Ð+ Ñ Ä B . ( More informally, “sequences are sufficient” to describe the topology in a first countable topological space.)

Proof (É ) Suppose Ð+Ñ88 is a sequence in E and that Ð+ÑÄB . For each neighborhood [ of BÐ+Ñ, 8 is eventually in [ . Therefore so [∩EÁg , so B−EÞ cl (This half of the proof works in any topological space: it does not depend on first countability.)

(ÊB−EÞ ) Suppose cl Using Lemma 9.4, choose a countable shrinking neighborhood base ÖY"8 ß ÞÞÞß Y ß ÞÞÞ× at BÞ Since B − cl E , we can choose a point + 88 − Y ∩ E for each 8 . By Lemma 9.5, Ð+8 Ñ Ä B . 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 HÐ\ßÑ is a dense subset in a first countable Hausdorff space g , then l\l Ÿ lHli! . In particular, If \ is a separable, first countable Hausdorff space, then i! l\l Ÿ i! œ -Þ

Proof \ßB−\Ð.ÑH is first countable so for each we can pick a sequence 8 in such that  Ð.8BB Ñ Ä B; formally, this sequence is a function 0 À Ä H , so 0 − H Þ Since \ is Hausdorff, a sequence cannot converge to two different points: so if BÁC−\ , then 0BC Á0 . Therefore the  i! function FFÀ\ÄH given by ÐBÑœ0B is one-to-one, so l\lŸlHlœlHl Þ ñ

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

Example 9.8 (the space P )

th Let P œ ÖÐ7ß 8Ñ À 7ß 8 −™ ß 7ß 8 !× , and let G4 be “the 4 column of P ,” that is G4 œ ÖÐ4ß 8Ñ − P À 8 œ !ß "ß ÞÞÞ×Þ We put a topology on P by giving a neighborhood base at each point :À

ÖÐ7ß 8Ñ×if : œ Ð7ß 8Ñ Á Ð!ß !Ñ U: œ  {F À Ð!ß !Ñ − F and C4  F is finite for all but finitely many 4× if : œ Ð!ß !Ñ

(Check that this definition satisfies the conditions in the Neighborhood Base Theorem 5.2 and

144 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 G+ß48! that contains infinitely many of the terms then

R œ ÐP  G48! Ñ ∪ ÖÐ!ß !Ñ× is a neighborhood of Ð!ß !Ñ and Ð+ Ñ is not eventually in R.

ii) if every column G+48 contains only finitely many 's, then RœPÖ+À8−88 × is a neighborhood of Ð!ß!Ñ and Ð+ ) is not eventually in RR (in fact, the sequence is never in ).

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 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 Ð ª Ñamong the basic neighborhoods at Ð!ß !Ñ is very complicated  much more complicated than the neat, simple nested chain of neighborhoods Y"# ª Y ª ÞÞÞ ª Y 8 ª ÞÞÞ that would 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

Sequences do suffice to describe the topology in a first countable space, so it is not surprising that we can use sequences to decide whether a function defined on a first countable space \ is continuous.

Theorem 9.9 Suppose Ð\ßggg Ñ is first countable and 0 À Ð\ß Ñ Ä Ð] ßw Ñ . Then 0 is continuous at + − \ iff whenever ÐB88 Ñ Ä + , then Ð0ÐB ÑÑ Ä 0Ð+ÑÞ

Proof (Ê0 ) If is continuous at +[ and is a neighborhood of 0Ð+Ñ0Ò[Ó , then " is a " neighborhood of + . Therefore ÐB88 Ñ is eventually in 0 Ò[ Ó , so 0ÐÐB ÑÑ is eventually in [ Þ (This half of the proof is valid in any topological space \Þ)

(É ) Let ÖY"8 ß ÞÞÞß Y ß ÞÞÞ× be a shrinking neighborhood base at + . If 0 is not continuous at +[0Ð+Ñ80ÒYÓ© , then there is a neighborhood of such that for every , 8 Î[ . " For each 8B−Y0Ò[ÓY , choose a point 88 . Then (since the 8 's are shrinking) we have ÐB88 Ñ Ä + but Ð0ÐB ÑÑ fails to converge to 0Ð+Ñ because 0ÐB 8 Ñ is never in [ . ñ ()Compare this to the proof of Theorem II.5.22.

145 10. Subsequences

Definition 10.1 Suppose0À9 Ä\ is a sequence in \ and that À Ä is strictly increasing. The composition 0‰9 À Ä\ is called a subsequence of 0 . 0 Ò\ 9ß9Å0‰ 

If we write 0Ð8Ñ œ B85 and 99 Ð5Ñ œ 8 , then Ð0 ‰ ÑÐ5Ñ œ 0Ð8 58 Ñ œ B5. We write the sequence 0 informally as ÐB88 Ñ and the subsequence 0 ‰99 as ( B5 Ñ . Since is increasing, we have that 8Ä∞5 as 5Ä∞Þ

For example, if 99Ð5Ñ œ 85 œ #5 , then 0 ‰ is the subsequence informally written as

ÐB85 Ñ œ ÐB #5 Ñ, that is, the subsequence ÐB #%' ß B ß B ß ÞÞÞß B #5 ß ÞÞÞÑÞ But if 9 Ð8Ñ œ 1 for all 8 , then 0 ‰9 is not a subsequence: informally, ÐB""" ß B ß B ß ÞÞÞß B " ß ÞÞÞ Ñ is not a subsequence of ÐB"#$ ß B ß B ß ÞÞÞß B 8 ß ÞÞÞ ÑÞ A sequence 0 is a subsequence of itself: just let 9 Ð8Ñ œ 8 .

Theorem 10.2 Suppose B − Ð\ßg Ñ . Then ÐB88 Ñ Ä B iff every subsequence ÐB5 Ñ Ä BÞ

Proof (ÉÐBÑ ) This is clear because 8 is a subsequence of itself.

(ÊÐBÑÄBÐBÑ ) Suppose 88 and that 5 is a subsequence. If [ is any neighborhood of BB−[88, then 8!55! whenever some . Since the 8's are strictly increasing, 88 for all

5 some 5!8 Þ Therefore ÐB55 Ñ is eventually in [ , so ÐB 8 Ñ Ä BÞ ñ

Definition 10.3 Suppose B − Ð\ßg Ñ Þ We say that B is a cluster point of the sequence ÐB8 Ñ if for each neighborhood [B of and for each 5 − , there is an 85for which B8 −[. More informally, we say that BÐBÑ is a cluster point of 8 if the sequence is frequently in every neighborhood [ of B . (The underlined phrases mean the same thing. )

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

Example 10.5.

1) Suppose \ œ Ò!ß "Ó ∪ Ö#× and E œ Ö#×Þ Then [ ∩ E Á g for every neighborhood of # , but # is not a limit point of E since [ ∩ ÐE  Ö#×Ñ œ gÞ Each B − Ò!ß "Ó is a limit point of Ò!ß "Ó and also a limit point of \ . Since Ö#× is open, # is not a limit point of \ .

2) In ‘ , every point

3) If ÐB888 Ñ Ä Bß then B is a cluster point of ÐB ÑÞ More generally, if ÐB Ñ has a

subsequence ÐB885 Ñ Ä B , then B is a cluster point of ÐB ÑÞ (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;8 ÑÞ

8 5) The only cluster points in ‘ for the sequence ÐB8 Ñ œ ÐÐ  "Ñ Ñ are  " and " . But the set ÖB8 À 8 −‘ × œ Ö  "ß "× 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 Ð\ßg Ñ is first countable and that + is a cluster point of ÐB8 Ñ . Then there is a subsequence ÐB85 Ñ Ä + .

Proof Let ÖY"# ß Y ß ÞÞÞß Y 8 ß ÞÞÞ× be a countable shrinking neighborhood base at +Þ Since ÐB8 Ñ is frequently in Y8B−YÐBÑY""8"8# , we can pick so that " . Since is frequently in , we can pick an

8#" 8 so that B 8# −Y #" ©YÞ Continue inductively: having chosen 8 " ÞÞÞ8 5 so that

B855 −Y ©... ©Y " , we can then choose 8 5"5 8 so that B 85" −Y 5"5 ©Y Þ Then ÐB 8 5 Ñ is a subsequence of ÐB88 Ñ and ÐB Ñ Ä +Þ ñ

Example 10.7 (the space P , revisited)

Let PÐBÑ be the space in Example 9.8 and let 8 be a sequence which lists all the elements of P  ÖÐ!ß !Ñ×Þ

Every basic neighborhood F of Ð!ß !Ñ is infinite, so F must contain terms B8 for arbitrarily large 8 . This means that ÐB88 Ñ is frequently in F , so Ð!ß !Ñ is a cluster point of ÐB ÑÞ

But no subsequence of ÐB8 Ñ can converge to Ð!ß !Ñ  because we showed earlier 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 ÐB88 Ñ Ä Ð!ß !Ñ in P . If there were infinitely many B Á Ð!ß !Ñ , 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 . Now let 0 À P Ä be any bijection. If ÐB8 Ñ Ä Ð!ß !Ñ in P , then 0ÐB88 Ñ œ 0ÐÐ!ß !ÑÑ œ 5 eventually, so Ð0ÐB ÑÑ Ä 5 œ 0ÐÐ!ß !ÑÑ in  .

The topology on  is discrete, so Ö5× is a neighborhood of 0Ð!ß !Ñ , but 0" ÒÖ5×Ó œ ÖÐ!ß !Ñ× is not a neighborhood of Ð!ß !Ñ . We conclude that 0 is not continuous at Ð!ß !Ñ even though Ð0ÐB88 ÑÑ Ä 0ÐÐ!ß !ÑÑ for every sequence ÐB Ñ Ä Ð!ß !ÑÞ

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 0ß1 À Ð\ßgg"# Ñ Ä Ð]ß Ñ are continuous functions, that H is dense in \ , and that 0lH œ 1lHÞ Prove that is ] is Hausdorff, then 0 œ 1Þ (This generalizes the result in Chapter 2, Theorem 5.12.)

E18. A function 0ÀÐ\ßg‘ ÑÄ is called lower semicontinuous if

0 "ÒÐ,ß∞ÑÓœÖBÀ0ÐBÑ,×is open for every ,−‘ , and 0 is called upper semicontinuous if

0 "[Ð∞ß,Ñ ] œÖBÀ0ÐBÑ,× is open for each ,−‘ .

a) Show that 00 is continuous iff is both upper and lower semicontinuous.

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

c) Prove that the characteristic function ;gE of a set E©Ð\ß Ñ 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. (Such a space ]X is called a " -space. Ñ Let 0À \ Ä ] be continuous and onto. Prove that either 0\ is constant or is homeomorphic to ] .

1) Note: the problem does not say that if 00 is not constant, then is a homeomorphism. 2) Hint: Prove first that if 0l\lœl]lÞ is not constant, then Then examine 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\ßgg Ñ and Ð] ßw Ñ are topological spaces. Recall that the product topology on \‚] is the topology for which the collection of “open boxes”

UggœÖY‚Z ÀY − ß Z −w × is a base.

a) The “projection maps” 11BCÀ\‚] Ä\ and À\‚] Ä] are defined by

11BCÐBß CÑ œ B and ÐBß CÑ œ CÞ

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

b) Suppose that Ð^ßg ww Ñ is a topological space and that 0 À ^ Ä \ ‚ ] Þ Prove that

148 0‰0À^ĉ0À^Ä] is continuous iff both compositions 11BC and are continuous.

c) Prove that ÐÐB88 ß C ÑÑ Ä ÐBß CÑ − \ ‚ ] iff ÐB 8 Ñ Ä B in \ and ÐC 8 Ñ Ä C 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\ßgf Ñ and ( ] ß Ñ be topological spaces. Suppose 0 À \ Ä ] . Let

>Ð0Ñ œ ÖÐBß CÑ − \ ‚ ] À C œ 0ÐBÑ× = “ the graph of 0 . ”

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

Note: if we think of 000 as a set of ordered pairs, the “graph of ” is . More informally, 29A/@/<ß the problem states that the graph of a function is homeomorphic to its domain iff the function is continuous.)

E23. In Ð\ßgYαA Ñ , a family of sets œ ÖFα À − × is called locally finite if each point B − \ has a neighborhood RR∩FÁg such that α 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 %%YY! such that .F (""# , F# ) for all F , F − . Prove that is locally finite.

b) Prove that if Yg is a locally finite family of sets in Ð\ß Ñ , then cl ( cl ( . Explain why this implies that if all the ' s are closed, αα−EFÑœαα −E FÑ Fα then is closed. ( This would apply, for example, to the sets in part a ) α−EFÞα c) (The Pasting Lemmas: compare Exercise II.E24 )

w Let Ð\ßgg Ñ and Ð] ß Ñ be topological spaces. For each α − E , suppose Fα © \ , that 0αα À F Ä ] is a continuous function, and that 0 αα lÐF ∩ F " Ñ œ 0 "α lÐF ∩ F " Ñ for all α" ß − E (that is, and agree wherever their domains overlap ). Then is a function and 00α" α−E 0œ0α (The function is built by “pasting together” all the “function pieces” . ) 0À Fα Ä]Þ 00α

i) Show that if all the F0α 's are open, then is continuous.

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

iii) Show that if there are only finitely many F0α 's and they are all closed, then

149 is continuous.(Hint: use a characterization of continuity in terms of closed sets.)

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

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

" L" À Ò!ß "Ó ‚ Ò!ß# Ó Ä Ð\ß .Ñ is continuous ß and " LÀÒ!ß"Ó‚Òß"ÓÄÐ\ß.Ñ# # is continuous, and "" LÐ>ßÑœLÐ>ßÑ"### 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) ÐÑor part c) says that the two functions can be pieced together into a continuous function # L ÀÒ!ß"Ó ÄÐ\ß.Ñß where L œL"# ∪L .

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 special properties assumed.

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

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

3. If B−Ð\ßg Ñ , then ÖB× is the intersection of a family of open sets.

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

w 5. Suppose gg and are topologies on \ and that for every subset E of \ , clgg ÐEÑ œ clw ÐEÑ . Then ggœÞw

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

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

8. If E©\ and H is dense in \ , then E∩H is dense in E .

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

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

11. Every space is a continuous image of a first countable space.

12. Let \ œ Ö!ß "× with the topology g œ Ögß \ß Ö 1 ×× . There are exactly 3 continuous functions 0 À Ð\ßgg Ñ Ä Ð\ß ÑÞ

13. 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.

14. If H©‘‘ and H is discrete in the subspace topology, then H is closed in .

151 15. If E©Ð\ßg Ñ and \ is separable, then E is separable.

16. Suppose ggg is the cofinite topology on \0ÀÐ\ßÑÄÐ\ßÑ . Then any bijection is a homeomorphism.

17. If XX has the cofinite topology, then the closure of every open set in is open.

18. IfÐ\ßggg Ñ is metrizable, and 0À Ð\ß Ñ Ä Ð] ßw Ñ is continuous and onto, then ( ]ßg w ) is also metrizable.

19. If DD is dense in \\ , then each point of is a limit of a sequence of points from .

20. A continuous bijection from ‘‘ to is a homeomorphism.

21. For every cardinal number 7Ð\ßl\lœ7 , there is a separable topological space g ) with .

22. If every family of disjoint open sets in Ð\ßgg Ñ is countable, then Ð\ß Ñ is separable.

## 23. For +−‘% and ! , let GÐ+ÑœÖB−% ‘ À .ÐBß+Ñœ % × . ## Let g‘ be the topology on for which the collection ÖG% Ð+ÑÀ %‘  !ß + − × is a subbase. Let h‘ be the usual topology on # . Then the function 0ÀÐßÑÄÐßÑ‘g## ‘h given by 0ÐBßCÑœB (sin , sin C ) is continuous.

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

25. Consider the property WÊ : “every minimal nonempty closed set is a singleton". Then T" S.

26. A one-to-one, continuous map0ÀÐ\ßgg ÑÄÐ]ß Ñ is a homeomorphism iff 0 is onto.

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

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

29. If DD is dense in Ð\ßgg Ñ and ‡‡ is topology on \ with gg © , then is dense in Ð\ß g ‡ Ñ .

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

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

32. Let U8 œ ÖF ©U À 8 − Fand F © { "ß #ß ÞÞÞß 8××Þ The 8 's satisfy the conditions in the Neighborhood Base Theorem and therefore describe a topology on  .

33. At each point 8−U , let 8 œÖFÀ F ªÖ8ß8"ß8#ßÞÞÞ××Þ The U8 's satisfy the conditions in the Neighborhood Base Theorem and therefore describe a topology on  .

34. Suppose Ð\ßgUU Ñ has a base with l l œ -Þ Then every open cover h of \ has a subcover h ww for which llŸ-h .

35. Suppose Ð\ßgUU Ñ has a base with l lœ-Þ Then \ has a dense set H with lHlŸ-Þ

36. Suppose Ð\ßgUU Ñ has a base with l l œ -Þ Then at each point B − \ , there is a neighborhood base with llŸ-UB .

37. If Ð\ßgUU Ñ has a base where l lœ7 is finite, then l g lŸ#7 .

38. Suppose \\ is an infinite set. Let gg"# be the cofinite topology on and let be the discrete topology on \Þ If 0 À‘g Ä Ð\ß"# Ñ is continuous, then 0 À ‘g Ä Ð\ß Ñ is also continuous.

39. Consider ‘g‘‘‘g with the topology œÖÐ+ß∞ÑÀ+− ×∪Ög , × . Ð ß Ñ is first countable.

153