Chapter II Metric and Pseudometric Spaces
1. Introduction
By itself, a set doesn't have any “structure.” For two arbitrary sets EFß and we can ask questions like “Is EœF ?” or “Is E equivalent to a subset of F ?” but not much more. If we add additional structure to a set, it becomes more interesting. For example, if we define a “multiplication operation” +†, in \ that satisfies certain axioms Ð such as +†Ð,†-ÑœÐ+†,ц-ÑÑ , then \ becomes an algebraic structure called a group and a whole area of mathematics known as “group theory” begins.
We are not interested in making a set \ into an algebraic system. For topology, we want additional structure on a set \\ for a different purpose: to talk about “nearness” in . This is what we need to discuss topics like “convergence” and “continuity”0 roughly, “ is continuous at + ” means that “if B is near + , then 0ÐBÑ is near 0Ð+ÑÞ ”
The simplest way to talk about “nearness” is to equip the set \. with a “distance function” to tell us “how far apart” two elements of \ are.
Note: As we proceed we may use ideas taken from elementary analysis, such as the continuity of a function 0À‘‘87 Ä as a source for motivation or examples ß although these ideas will not be carefully defined until later in this chapterÞ
2. Metric and Pseudometric Spaces
Definition 2.1 Suppose .À\‚\Ä‘ and that for all BßCßD−\ :
1) .ÐBß CÑ ! 2) .ÐBß BÑ œ ! 3) .ÐBß CÑ œ .ÐCß BÑ (symmetry ) 4) .ÐBß DÑ Ÿ .ÐBß CÑ .ÐCß DÑ ( the triangle inequality )
Such a “distance function” .Ð\ß.Ñ is called a pseudometric on \ . The pair is called a pseudometric space. If . also satisfies
5) when BÁC , then .ÐBßCÑ! then .\\ß.Ñ is called a metric on and ( is called a metric space . Of course, every metric space is automatically a pseudometric space.
If a pseudometric space Ð\ß .Ñ is not a metric space ßit is because there are at least two points B Á C for which .ÐBß CÑ œ !Þ In most situations this doesn't happen; metrics come up in mathematics more often than pseudometrics. However pseudometrics do occasionally arise in a natural way. Moreover, many definitions and theorems actually only require using properties 1)-4). Therefore we will state our results in terms of pseudometrics rather than metrics in situations where 5) is irrelevant.
61 Example 2.2
1) The usual metric on ‘ is .ÐBß CÑ œ lB ClÞ Clearly, properties 1) - 5) are true. In fact, 1)-5) are chosen so that a metric imitates the usual distance function.
8 2) The usual metric on ‘ is defined as follows: if B œ ÐB"# ß B ß ÞÞÞß B 8 Ñ and 8 are in 8 then # . You should already know that has C œ ÐC"# ß C ß ÞÞÞß C 8 Ñ‘ ß .ÐBß CÑ œ ÐB 3 C 3 Ñ . 3œ" properties 1) - 5). But the details for verifying the triangle inequality are a little tricky, so we will go through the steps. First, we prove another useful inequality.
8 Suppose E œ Ð+"# ß + ß ÞÞÞß + 8 Ñ and F œ Ð, "# ß , ß ÞÞÞß , 8 Ñ points in ‘ Þ Define 8888 ## ## T ÐAÑ œ Ð+33 A, Ñ œ +33 Ð# + 33 , ÑA Ð , ÑA 3œ" 3œ" 3œ" 3œ" T ÐAÑ is a quadratic function of Aß and T ÐAÑ ! because T ÐAÑ is a sum of squares. Therefore the equation TÐAÑœ ! has at most one real root, so it follows from the quadratic formula that
888 ### Ð# +33 , Ñ %Ð +33 Ñ Ð , Ñ Ÿ ! , which gives 3œ" 3œ" 3œ" 888 # "Î# # "Î# |+,lŸÐ33 +Ñ33 Ð , Ñ 3œ" 3œ" 3œ"
This last inequality is called the Cauchy-Schwarz inequality . In vector notation it could be written in the form lE † Fl Ÿ llEll † llFllÞ
8 Then if B œ ÐB"# ß B ß ÞÞÞß B 8 Ñ , C œ ÐC "# ß C ß ÞÞÞß C 8 Ñ and D œ ÐD "# ß D ß ÞÞÞß D 8 Ñ are in ‘ ß we can calculate 88 ## # .ÐBß DÑ œ ÐB33 D Ñ œ ÐÐB 33 C Ñ ÐC 33 D ÑÑ 3œ" 3œ"
88 8 ## œ ÐB33 CÑ # ÐB 3333 CÑÐC DÑ ÐC 33 DÑ 3œ" 3œ" 3œ"
88 8 ## Ÿ ÐB33 CÑ # lB 3333 CllC Dl ÐC 33 DÑ 3œ" 3œ" 3œ"
888 8 # # "Î# # "Î# # Ÿ ÐBCÑ#Ð33 ÐBCÑÑ 33 Ð ÐBCÑÑ 33 ÐCDÑ 33 3œ" 3œ" 3œ" 3œ"
œ Ð.ÐBß CÑ .ÐCß DÑÑ# . Taking the square root of both sides gives
.ÐBß DÑ Ÿ .ÐBß CÑ .ÐCß DÑ .
Example 2.3 We can also put other “unusual” metrics on the set ‘8Þ
62 1) Let . be the usual metric on ‘8w and define . ÐBß CÑ œ "!!.ÐBß CÑ . Then . w is also a metric on ‘‘88wÞÐß.Ñ In , the “usual” distances are stretched by a factor of 100. It is as if we simply changed the units of measurement from meters to centimeters and that change shouldn't matter in any important way. In fact, it's easy to check that if .\ is any metric (or pseudometric) on a set and αα!, then .w œ †. is also a metric (or pseudometric) on \ .
8 2) If B œ ÐB"# ß B ß ÞÞÞß B 8 Ñ , C œ ÐC "# ß C ß ÞÞÞß C 8 Ñ are points in ‘ ß define
8 .ÐBßCÑœ>33 lBCl 3œ"
8 It is easy to check that .Ðß.Ñ.>>> satisfies properties 1) - 5) so ‘ is a metric space. We call the taxicab metric on ‘8Þ ( For 8œ# , distances are measured as if you had to move along a rectangular grid of city streets from BC to the taxicab cannot cut diagonally across a city block ).
8 3) If B œ ÐB"# ß B ß ÞÞÞß B 8 Ñ , C œ ÐC "# ß C ß ÞÞÞß C 8 Ñ are points in ‘ ß define
‡ . ÐBß CÑ œ max ÖlB33 C l À 3 œ "ß #ß ÞÞÞß 8×
Then Ðß.Ñ‘‘8‡ is also a metric space. We will refer to . ‡ as the max metric on 8 Þ
‡ When 8œ"ß of course, .ß.> and . are exactly the same metric on ‘ .
w‡ 8 We will see later that ..ß. ,> are all “equivalent” metrics on ‘ for “topological” purposes. Roughly, this means that whichever of these metrics is used in ‘8ß exactly the same functions are continuous and exactly the same sequences converge.
63 4) The “unit sphere” W"# is the set of points in ‘ that are at distance 1 from the origin. Sketch #‡w the unit sphere in ‘ using the metrics .ß .> ß . , and . œ "!!.Þ Since there are only two coordinates, we will write a point in ‘# in the usual way as ÐBß CÑ rather than ÐB"# ß B Ñ .
For ..ß , we get For > we get
"" W œ ÖÐBß CÑ À .Ð ÐBß CÑß Ð!ß !ÑÑ œ "× W œ ÖÐBß CÑ À .> Ð ÐBß CÑß Ð!ß !ÑÑ œ "× œÖÐBßCÑÀB## C œ"× œÖÐBßCÑÀlBllClœ"×
For .‡ , we get
W"‡ œ ÖÐBß CÑ À . Ð ÐBß CÑß Ð!ß !ÑÑ œ "× œ ÖÐBß CÑ Àmax ÖlBlß lCl× œ "×
Of course for the metric .w" œ "!!. , W has the same shape as for the metric . , but the sphere is reduced in size by a scaling factor of "!! .
w‡ 8 Switching among the metrics .ß . , .> ß . produces unit spheres in ‘ with different sizes and shapes . In other words, changing the metric on ‘8 may cause dramatic changes in the geometry of the space for example, “areas” may change and “spheres” may no longer be “round.” Changing the metric can also affect smoothness features of the space ÐÑ spheres may turn out to have sharp corners . But it turns w‡ out, as mentioned earlier, that .ß . , .> and . are “equivalent” for “topological purposes.” For topology, “size,” “geometrical shape,” and “smoothness” don't matter.
64 When talking about ‘‘8ß. the usual metric is the default that is, we always assume that 8 , or any subset of ‘8 , has the usual metric . unless a different metric is explicitly stated.
Example 2.4 In each part, you should verify that . satisfies the properties of a pseudometric or metric.
1) For a set \ , define .ÐBßCÑœ! for all BßC−\Þ We call . the trivial pseudometric on \ : all distances are 0. (Under what circumstances is this . a metric? )
!BœCif 2) For a set \.ÐBßCÑœ , define . We call . the discrete unit metric on \ . "BÁCif To verify the triangle inequality: for points Bß Cß D − \ , .ÐBß DÑ Ÿ .ÐBß CÑ .ÐCß DÑ certainly is true if BœD; and if BÁD , then .ÐBßDÑœ" and .ÐBßCÑ.ÐCßDÑ " .
Definition 2.5 Suppose Ð\ß .Ñ is a pseudometric space, that B! − \ and % ! . Then FÐBÑœÖB−\À.ÐBßBÑ% !!% × is called the ball of radius % with center at B ! . If there exists an % ! such that FÐBÑœÖB×% !! , then we say that B ! is an isolated point in (\ß.ÑÞ
Example 2.6
8 1) In ‘%%‘ , FÐBÑœÐB%%!!! ßB Ñ . More generally, FÐBÑ ! in is just the usual spherical 8 ball with radius % and center at BÞ.ß!> (not including the boundary surface) If the metric is used in ‘ thenFÐBÑ% ! is the interior of a “diamond-shaped” region centered at B! . ÐSee the earlier sketches of "# " W: in Б , .>" Ñ , F ÐÐ!ß !ÑÑ is the region “inside” the diamond-shaped W ÞÑ " In \ œ Ò!ß "Ó with the usual metric . , then F" Ð!Ñ œ Ò!ß Ñ , F"# Ð!Ñ œ Ò!ß "Ñ ß F Ð!Ñ œ Ò!ß "ÓÞ # #
2) If . is the trivial pseudometric on \ and B!! −\ß then FÐBÑœ\% for every % !Þ
ÖB! ×if % Ÿ " $Ñ If . is the discrete unit metric on \ , then F% ÐB! Ñ œ . Therefore \"if % every point BÐ\ß.Ñ! in is isolated. The same is true if we rescale and replace . by the metric α . Ð!Ñwhere α .
4) Let GÐÒ!ß "ÓÑ œ Ö0 −‘Ò!ß"Ó À 0 is continuous. × For 0, 1 − GÐÒ!ß "ÓÑß define
.Ð0ß1Ñ œ" l0ÐBÑ1ÐBÑl.B (*) ! It is easy to check that . is a pseudometric on GÐÒ!ß "ÓÑ . In fact . is a metric: if 0 Á 1 , then there must be a pointB!!! − Ò!ß "Ó where l0ÐB Ñ 1ÐB Ñl ! . By continuity, l0ÐBÑ 1ÐBÑl ! for B 's near B! , that is, l0ÐBÑ 1ÐBÑl ! on some interval Ò+ß ,Ó © Ò!ß "Óß where B! − Ò+ß ,ÓÞ (carefully explain why! ). Let 7œ minB−Ò+ß,Ó l0ÐBÑ1ÐBÑl (why does 7 exist? ). Then 7! , so
.Ð0ß 1Ñ œ",, l0ÐBÑ 1ÐBÑl .B l0ÐBÑ 1ÐBÑl .B 7 .B œ 7Ð, +Ñ !Þ !++ Therefore, . is a metric on GÐÒ!ß "ÓÑÞ
65 GÐÒ!ß "ÓÑis a subset of the larger set ] œ Ö0 −‘Ò!ß"Ó À 0 is integrable × . We can define a distance function .] on the same formula (*). In this case, . is a pseudometric on ] but not a metric. If
" !BÁif # 0ÐBÑ œ ! for all B , and 1ÐBÑ œ" , then 0 Á 1 but "Bœif # .Ð0ß1Ñ œ" l0ÐBÑ1ÐBÑl.B œ ! ! This example shows how a pseudometric that is not a metric can arise naturally in analysis.
5) On GÐÒ!ß "ÓÑ we can also define another metric .‡ by
.‡ Ð0ß1Ñ œ sup Ö0ÐBÑ1ÐBÑl | À B − Ò!ß"Ó× œ max| Ö 0ÐBÑ 1ÐBÑl À B − Ò!ß "Ó×
(Replacing “sup” with “max” makes sense because a theorem from analysis says that the continuous function l0 1lhas a maximum value on the closed interval Ò!ß "Ó.)
‡ Then . Ð0ß1Ñ %% if and only if l0ÐBÑ1ÐBÑl at every point B − Ò!ß"Ó , so we can picture F% Ð0Ñ in ÐGÐÒ!ß "ÓÑß .‡ Ñ as the set of all functions 1 − GÐÒ!ß "ÓÑ whose graph lies entirely inside a “tube of width %%” containing the graph of 0 that is, 1−FÐ0Ñ% iff 1 is “uniformly within of 0 on Ò!ß"Ó .” See the following figure.
How are the metrics . and .‡ from Examples 4) and 5) related? Notice that for 0ß 1 − GÐÒ!ß "ÓÑ :
.Ð0ß 1Ñ œ"" l0ÐBÑ 1ÐBÑl .B Ÿmax l0ÐBÑ 1ÐBÑl .B œ " .‡‡ Ð0ß 1Ñ .B œ . Ð0ß 1Ñ Þ !!B−Ò!ß"Ó !
‡..‡ We abbreviate this observation by writing .Ÿ.Þ It follows that F%% Ð0Ñ©F Ð0Ñ : so, for a given % !ß the larger metric produces the smaller ball. (Note: the superscript notation on the balls indicates which metric is being used in each case.)
..‡ The following below shows a function 01−FÐ0ÑFÐ0 and the graph of a function %% . The graph of10 coincides with the graph of , except for a tall spike: the spike takes the graph of 1 outside the “% - tube” around the graph of 0 , but the spike is so thin that the .Ð0ß 1Ñ œ" l0ÐBÑ 1ÐBÑl .B ! œ01“the total area between the graphs of and ”% .
66
∞ # 6) Let jœÖ0−# ‘ À 0Ð5Ñ converges × . If we write 0Ð5ÑœB5 and use the more informal 5œ" ∞ # sequence notation, then jœÖÐBÑÀB−#55‘ and B5 converges × . Thus, j # is the set of all “square- 5œ" summable” sequences of real numbers.
Suppose BœÐBÑ55# and CœÐCÑ are in j and that +ß,−‘ Þ We claim that the sequence +B ,C œ Ð+B55 ,C Ñ is also in j # Þ To see this, look at partial sums:
8888888 ### #### # Ð+B5 ,CÑ 5 œ+ B555 #+, B 55 C , C Ÿ+ B l#+, B 55 C l, C 5 5œ" 5œ" 5œ" 5œ" 5œ" 5œ" 5œ"
8888 # # # "Î# # "Î# # Ÿ + B555 #l+ll,lÐ B Ñ Ð C Ñ , C5 ()by the Cauchy-Schwarz inequality 5œ" 5œ" 5œ" 5œ"
∞∞∞∞ # # # "Î# # "Î# # Ÿ + B555 #l+ll,lÐ B Ñ Ð C Ñ , C5 œ Q − ‘ (all the series converge 5œ" 5œ" 5œ" 5œ" because Bß C − j# ÞÑ
∞ # Therefore the nonnegative series Ð+B55 ,C Ñ converges because it has bounded partial 5œ" sums. This means that +B ,C − j# Þ
∞ # In particular, if BßC−jß##55 we now know that BC−j so ÐB CÑ converges. Therefore it 5œ" ∞ # "Î# makes sense to define .ÐBß CÑ œ Ð ÐB55 C Ñ Ñ Þ You should check that . is a metric on j # . 5œ" 888 # "Î# # "Î# # "Î# (For the triangle inequality, notice that ÐÐBDÑÑŸÐÐBCÑÑÐÐCDÑÑ55 55 55 5œ" 5œ" 5œ" 8 by the triangle inequality in ‘ . Letting 8Ä∞ gives the triangle inequality for j# .)
67 7) Suppose Ð\33 ß . Ñ are pseudometric spaces Ð3 œ "ß ÞÞÞß 8Ñ , and that B œ ÐB"8 ß ÞÞÞß B Ñ and C œ ÐC"8 ß ÞÞÞß C Ñ are points in the product \ œ \ " ‚ ÞÞÞ ‚ \ 8 Þ Then each of the following is a pseudometric on \À 88 # "Î# .ÐBßCќР.3 ÐBßCÑÑ33 .ÐBßCÑœ > .ÐBßCÑ 333 3œ" 3œ"
‡ . ÐBß CÑ œ max Ö.333 ÐB ß C Ñ À 3 œ "ß ÞÞÞß 8×
‡ If each ..ß.ß.Þ\œ.3> is a metric, then so are and Notice that if each 33‘ and each is the usual ‡ 8 metric on ‘ , then .ß .> ß and . are just the usual metric, the taxicab metric, and the max metric on ‘ Þ As we shall see, it turns out that these metrics on \ are all equivalent for “topological purposes.”
Definition 2.7 Suppose Ð\ß .Ñ is a pseudometric space and S © \ . We say that S is open in Ð\ß .Ñ if for each B−S there is an % ! such that FÐBÑ©S% . (Of course, % may depend on B . )
For example, i) The sets g and \ are open in any space Ð\ß .ÑÞ
ii) The intervals Ð+ß ,Ñ , Ð ∞ß +Ñß Ð,ß ∞Ñß and Ð ∞ß ∞Ñ œ ‘‘ are open in . (Fortunately this terminology is consistent with the fact that these intervals were called “open intervals” in calculus books. ) But notice that the interval Ð+ß ,Ñ , when viewed as a subset of the B -axis in ‘# , is not open in ‘‘# . Similarly, is an open set in ‘‘, but (viewed as the B -axis) is not open in ‘# .
iii) The intervals Ò+ß ,Óß Ò+ß ,Ñ and Ð+ß ,Ó are not open in ‘ Þ But the sets Ò+ß ,Ñ and Ò+ß ,Ó are open in the metric space ÐÒ+ß ,Óß .Ñ .
Examples ii) and iii) illustrate that “open” is not an a property that depends just on the set E : whether or not a set E is open depends on the larger space in which it is contained that is, “open” is a relative term.
The next theorem tells us that the balls in Ð\ß .Ñ are the “building blocks” from which all open sets can be constructed.
Theorem 2.8 A set S©\ is open in Ð\ß.Ñ if and only if S is a union of a collection of balls.
Proof If S is open, then for each B−S there is an %B ! such that F%B ÐBÑ©S and Sœ F ÐBÑÞ B−S %B Conversely, suppose Sœ F ÐBÑ for some indexing set G©SÞ We must show that if B−G %B C−S, then FÐCÑ©S for some % !Þ Since C−Sß we know that C−F ÐBÑ for some B −GÞ % %B! ! ! " Then .ÐB!B ß CÑ œ$% !! . Let % œ# Ð % B $ Ñ ! and consider F%% ÐCÑÞ If D − F ÐCÑ , then """"" .ÐDßBÑŸ.ÐDßCÑ.ÐCßBÑ!!BBBBB%$%$$% œ##### Ð!!!!! Ñ œ $% % œ % ß so D−F ÐBÑÞ Therefore FÐCÑ©F ÐBÑ©S . ñ %%%BB!!!!
Corollary 2.9 a) Each ball FÐBÑ% is open in Ð\ß.Ñ . b) A point BÐ\ß.ÑÖB×!! in a pseudometric space is isolated iff is an open set.
68 Definition 2.10 Suppose Ð\ß .Ñ is a pseudometric space. The topology g. generated by . is the collection of all open sets in Ð\ß .Ñ . In other words, g. œ ÖS À S is open in Ð\ß .Ñ× œ ÖS À S is a union of balls×Þ
Theorem 2.11 Let g. be the topology in Ð\ß .ÑÞ Then
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
(Conditions ii) and iii) say that the collection g. is “closed under unions” and “closed under finite intersections.” )
Proof is the union of the empty collection of open balls, and , so . g\œFÐBÑgß\−B−\ ".g Suppose where each . Then is in one of these open sets, say B−Sœα−E Sαα S −g. B Sα%α!!. So for some %g !ßB−FÐBÑ©S ©S . Therefore S is open, that is, S−. Þ
To verify iii), suppose S"# ß S ß ÞÞÞß S 8 −g . and that B − S " ∩ S # ∩ ÞÞÞ ∩ S 8 . For each
3 œ "ß ÞÞÞß 8ßthere is an %%%%%33"#8 ! such that B − F%3 ÐBÑ © S . Let œ min Ö ß ß ÞÞÞß × ! . Then FÐBÑ©S% "# ∩S ∩ÞÞÞ∩S 8. Therefore S "# ∩S ∩ÞÞÞ∩S 8. −g Þ ñ
Example 2.12 The set SœÐßÑ"" is open in ‘ for every 8− . However, ∞ SœÖ!× is 8 88 8œ" 8 not open in ‘ : so an intersection of infinitely many open sets might not be open. (Where does the proof for part iii) in Theorem 2.11 break down if we intersect infinitely many open sets? )
Notice that different pseudometrics can produce the same topology on a set \Þ For example, if . is a metric on \.œ#... and we set ww , then and produce the same collection of balls (with radii ..w measured differently): for each % ! , the ball F% ÐBÑ is the same set as the ball F#% ÐBÑ . If we get the same balls from each metric, then we must also get the same open sets: gg..œ w (see Theorem 2.8 ).
#‡ We can see a less trivial example in ‘ . Let .ß .> ß and . be the usual metric, the taxicab metric, and the max metric on ‘#‡ . Clearly any set which is a union of .. -balls (or balls) can also be written as a # union of .> -balls, and vice-versa. (Explain why! See the following picture for ‘ . )
69
‡ Therefore all three metrics produce the same topology: gg..œœ> g . even though the balls are different for each metric. It turns out that the open sets in Ð\ß .Ñ are the most important objects from a “topological” point of view, so in that sense these metrics are all equivalent. (As mentioned above, these metrics do change the “shape” and “smoothness” of the balls and therefore these metrics are not equivalent “for geometric purposes.”)
Definition 2.13 Suppose .. and w are two pseudometrics (or metrics) on a set \ . We say that . and ww w ..µ.Ñœ.. are equivalent (written if gg..w , that is, if and generate the same collection of open sets.
Example 2.14
1) If .\ÖBלFÐBÑ is the discrete unit metric on , then each singleton set " is a ball, so each is open equivalently, every point is isolated in If , then is open ÖB× B Ð\ß .ÑÞ E © \ E œ+−E Ö+× because EœÐ\Ñ is a union of balls. Therefore gc. , called the discrete topology on \
"(if B Á C If .ÐBßCÑœww on a set \ , then . µ. , where . is the discrete unit metric. !BœCif
More generally, αα.µ. for any !Þ , and all of these metrics generate the discrete topology.
" w 2) Let \œÖ8 À8− × . Let . be the usual metric on \ and let . be the discrete unit . "" metric on \Þ For each 8 , F% Ð88 Ñ œ Ö × if we choose a sufficiently small % . Therefore, just as in part 1), every subset of \ is open in Ð\ß.ÑÞ But every subset in Ð\ß.ww Ñ also is open, so . µ . (even though .. and w are not constant multiples of each other).
3) If .\.\ is the trivial pseudometric on a set , are there any other pseudometrics w on for which .µ.w ?
3. The topology of ‘
What do the open sets in ‘%‘%% look like? Since an -ball in is an interval of the form (+ ß+ Ñ , the open sets are precisely the sets which are unions of open intervals. But we can say more to make the situation even clearer. We begin by making precise the definition of “interval.”
70 Definition 3.1 A subset I of ‘ is convex if whenever BŸCŸD and BßD−M , then C−M . A convex subset of ‘ is called an interval .
It is easy to give examples of intervals in ‘ . The following theorem states that the obvious examples are the only examples.
Theorem 3.2 M©‘ is an interval iff M has one of the following forms (where +, ):
( ∞ß ∞Ñ , ( ∞ß +Ñ , ( ∞ß +Ó , Ò+ , ∞Ñ , Ð+ , ∞Ñ , Ð+,b ), Ò+ß ,Ñß Ð+ß ,Ó , Ò+ß ,Óß Ö+× , g (*)
Proof It is clear that each of the sets in the list is convex and therefore is an interval.
Conversely, we need to show that every interval MlMlŸ"ß has one of these forms. Clearly, if then Mœg or MœÖ+×Þ If lMl # then the definition of interval implies that M must be infinite.
The remainder of the proof uses the completeness property (œÑ “least upper bound property” of ‘ , and the argument falls into several cases:
Case I: MM is bounded both above and below. Then has a least upper bound and a greatest lower bound: let +œ inf M and ,œ sup MÞ Of course + and , may or may not be in M .
a) if +ß , − M, we claim M œ Ò+ß ,Ó b) if +−Mbut ,ÂM, we claim MœÒ+ß,Ñ c) if +ÂM but ,−M , we claim MœÐ+ß,Ó d) if,+ß ,  M we claim M œ Ð+ß ,Ñ
Case II: M+œMÞ is bounded below but not above. In this case, let inf a) if +−M, we claim MœÒ+ß∞Ñ b) if +ÂMß we claim MœÐ+ß∞Ñ
Case III: M,œMÞ is bounded above but not below. In this case, let sup a) if ,−M , we claim MœÐ∞ß,Ó b) if ,ÂM, we claim MœÐ∞ß,Ñ
Case IV: MMœÐ∞ß∞ÑÞ is not bounded above or below. In this case, we claim
The proof is similar in each case, using properties of sups and infs. To illustrate, We prove case Ic):
If B−Mß then BŸsup Mœ,Þ Also, B inf Mœ+ß and because +ÂMßwe get B+. So M©Ð+ß,ÓÞ
We need to show that M©Ð+ß,Ó , so suppose B−Ð+ß,Ó . Then B+œ inf Mß so BM is not a lower bound for . This means that there is a point D−M such that DBÞ Then DBŸ, where Dß,−M , and M is convex, so B−MÞ Therefore Ð+ß ,Ó © M , so M œ Ð+ß ,ÓÞ ñ
71 Note: We used the completeness property to prove Theorem 3.2. In fact, Theorem 3.2 is equivalent to the completeness property. To see this À
Assume Theorem 3.2 is true and that E is a nonempty subset of ‘ that has an upper bound. Let M œ ÖB −‘ À B Ÿ - for some - − E×Þ Then M is an interval Ð suppose BŸCŸD where BßD−M . Then z Ÿ- for some -−E ; therefore CŸ-=9 , by definition of MC−MÑÞ, .
Since EÁgßM must be infinite. An upper bound for E must also be an upper bound for M . Since I is an interval with an upper bound, I must have one of the forms ( ∞ß ,Ó , ( ∞ß ,Ñß Ð+ ,b ), Ò+ß ,Ñß Ð+ß ,Ó , Ò+ß ,Ó , Then it's not hard to check that EEœ,Þ has a least upper bound, namely sup
This is an observation I owe to Professor Robert McDowellÞ
It is clear that an intersection of intervals in ‘ is an interval (why ?). Also, a union of intervals need not be an interval: for example Ò!ß "Ó ∪ Ò#ß $Ó . But if every pair of intervals in a collection “overlap,” then the union is an interval. The following theorem makes this precise.
Theorem 3.3 Suppose is a collection of intervals in . If for all , then is \‘\\M∩NÁg MßN− an interval. In particular, if , then is an interval. \\Ág Proof Let . Then there are intervals in with and . +ß,−\\ MßN +−M ,−N Suppose +ŸBŸ,.
Pick a point . If , then and we are done. Otherwise, either D−M∩N BœD B− \ DBŸ, or +ŸBDÞ Therefore either B is either between two points of N and so B−N ; or B is between two points of MB−M , so . Either way, we conclude that
B− \ Þ ñ
We can now give a more careful description of the open sets in ‘ .
Theorem 3.4 Suppose S©‘‘ . OO is open in if and only if is the union of a countable collection of pairwise disjoint open intervals.
Proof (É ) Open intervals in ‘ are open sets, and a union of any collection of open sets is open.
(ÊS ) Suppose is open in ‘ . For each B−S , there is an open interval (ball) M such that . Let is an open interval and . Then . B−M©S KB œ ÖMÀM B−M©S× B−KB ©S By Theorem 3.3, KB is also an open interval (in fact, KBB is the largest open interval containing and inside S; why? ). It is easy to see that there can be distinct points Bß C − S for which KBC œ K . In fact, we claim that if Bß C − S , then either KBC œ K or K BC ∩ K œ g .
If K∩KÁgBC , then there is a point D−K∩K BC . By Theorem 3.3, K∪K BC is an open interval, a subset of SBCK∪K , and containing both and . Therefore BC is a set in the collection whose union is KK∪K©KBBC . Therefore B . Similarly, K∪K©KBC C, so KœKÞ B C
72 Removing any repetitions, we let W be the collection of the distinct intervalsKB that arise in this way. Clearly, Sœ ∪WW and is countable because the members of W are pairwise disjoint: for each M−W, we can pick a rational number ;MM −M , and these ; 's are distinct. (More formally, the function 0ÀW Ägiven by 0ÐMÑœ;M is one-to-one.) ñ
We are now able to find the number of open sets in ‘ .
Corollary 3.5 There are exactly c. open sets in ‘
Proof Let g‘g.. be the usual topology on . We want to prove that llœ-Þ
For each <−‘gg , the interval Ð∞ß<Ñ−.. , so l l -Þ
Let \‘\g be the set of all open intervals in . Then llœ- (why? Ñ . For each S−. , pick a sequence for which ∞ . (We could also choose the 's to be pairwise disjoint, M"# ß M ß ÞÞÞM 8 ß ÞÞÞ −\ S œ8œ" M 8 M8 but that is unnecessary in this argument the important thing here is that there are only countably many M8's.) Then we have a function 0 Àg\."#8 Ä given by 0ÐSÑ œ ÐM ß M ß ÞÞÞß M ß ÞÞÞÑ . The function 0 is clearly one-to-one, so lg\ lŸl lœ-i! œ-Þ ñ
73 Exercises
E1ÞÐ\ß.Ñ The following statements refer to a metric space . Prove the true statements and give counterexamples for the false ones. (The statements illustrate the danger of assuming that familiar features of ‘8 necessarily carry over to arbitrary pseudometric spaces. )
a) FÐBÑœ%% FÐCÑ implies BœC (i.e., “a ball can't have two centers”) b) The diameter of FÐBÑ% must be bigger than % . ÐThe diameter of a set A in a metric space is defined to be sup Ö.ÐBß CÑ À Bß C − E× Ÿ ∞ . Ñ
# E2ÞBC The “taxicab” metric on ‘ is defined by d>""##ÐÐB ß C Ñß ÐB ß C ÑÑ œ | " B # | | " C # |. Draw # the set of points in Б ,d> Ñ that are equidistant from Ð!ß !Ñ and Ð$ß %ÑÞ
E3. Suppose Ð\ß .Ñ is a metric space Þ
+Ñ Define .‡‡ ( Bß CÑ œ min Ö"ß .ÐBß CÑ× . Prove that . is also a metric on \ , and that gg..œ ‡ . .ÐBßCÑ ‡‡ ‡‡ b) Define .BßCÑœ (".ÐBßCÑ . Prove that . is also a metric on \ and that gg.. œ ‡‡ Hint: Let .\0 be a metric on and suppose is a function from the nonnegative real numbers to the nonnegative real numbers for which: 0Ð!Ñœ!ß BŸCÊ0ÐBÑŸ0ÐCÑ , and 0ÐB CÑ Ÿ 0ÐBÑ 0ÐCÑ for all nonnegative Bß CÞ Prove that .w ÐBß CÑ œ 0Ð.ÐBß CÑÑ is also a metric on B \Þ Then consider the particular function 0ÐBÑ œ"B Þ
Note: For all Bß C in \ , d‡‡‡‡‡‡ ( Bß CÑ Ÿ " and . ÐBß CÑ Ÿ " that is, d and . are bounded metrics on \\. Thus any metric d on can be replaced by an equivalent bounded metric that is, a bounded metric generating the same topology. So “boundedness” is a property determined by the particular metric, not the topology.
E4. Suppose 0À‘‘ Ä . Let . be the usual metric on ‘ , and .w# the usual metric on ‘ . Define a new distance function .ww on ‘ by . ww ÐBß CÑ œ . w Ð ÐBß 0ÐBÑÑß ÐCß 0ÐCÑÑÑ . Prove that . ww is a metric on ‘. Must ..ww be equivalent to ? If not, can you describe conditions which will guarantee that .µ.ww ?
E5. Suppose a function .À\‚\Ä‘ satisfies the conditions 1), 2), 3), and 5) in the definition of a metric, but that instead of the triangle inequality, we have that for all Bß Cß D − \
.ÐBß DÑ .ÐBß CÑ .ÐCß DÑÞ Prove that l\l Ÿ "Þ
E6. Suppose EÐ\ß.ÑE is a finite open set in a metric space . Prove that every point of is isolated in Ð\ß .ÑÞ
E7. Suppose Ð\ß .Ñ is a metric space and B − \Þ Prove that the following two statements are equivalent: i) B\ is not an isolated point of ii) every open set containing B contains an infinite number of points.
74 E8. The definition of an open set in Ð\ß .Ñ reads: S is open if for all B − Sß there is an % ! such that FÐBÑ©S% . In this definition, % may depend on B . Suppose we define S!B−S to be uniformly open if there is an % such that for all , FÐBÑ©S% that is, the same % works for every B−SÞÐ “Uniformly open” is not a standard term.Ñ
a) What are the uniformly open subsets of ‘8 ? b) What are the uniformly open sets in Ð\ß .Ñ if . is the trivial pseudometric? c) What are the uniformly open sets in Ð\ß .Ñ if . is the discrete unit metric?
E9. Let p be a fixed prime number. We define the :-adic absolute value | |: (sometimes called the : -adic norm) on the set of rational numbers as follows:
pmk If 0ÁB− , write Bœ n for integers k,m,n , where p does not divide m or n, and define 5 " ||Bœ:œ: :5 . (Of course, 5 may be negative. ). Also, define || !œ!Þ:
Prove that ll: “behaves the way an absolute value (norm) should” that is, for all BßC−
a) lBl:: ! and lBl œ! iff Bœ! b) l BCl::: œ l Bl † l Cl c) lBCl::: ŸlBl lCl ll: actually satisfies a stronger inequality than the inequality in part c). Prove that
d) lBCl::::: Ÿmax ÖlBl ß lCl ן lBl lCl
Whenever we have an absolute value (norm), we can use it to define a distance function:
for Bß C − ß let .:: ÐBß CÑ œ lB Cl
e) Prove that ..:: is a metric on , and show that actually satisfies an inequality stronger than the usual triangle inequality, namely:
for all Bß Cß D −, .::: ÐBß DÑ Ÿmax Ö. ÐBß CÑß . ÐCß DÑ×
f) Give a specific example for Bß Cß Dß : for which
.ÐBßDÑ::: max Ö.ÐBßCÑß.ÐCßD×Ñ
(Hint: It might be convenient to be able to refer to the exponent “5B ” associated with a particular . If pmk Bœ n , then 5:B roughly refers to the “number of 's that can be factored out of ” so we can call 5 œ// ÐBÑÞProve that (a b) min Ö // Ð+Ñß Ð,Ñ× whenever a,b − with a,b Á 0 and a Á b. Note that strict inequality can occur here: for example, when pœœœ 3, // (8) (2) 0, but //(8œ 2) (6) œ 1. )
888 g) Suppose : œ #Þ Calculate .### Ð# ß !ÑÞ What are lim . Ð# ß !Ñ and lim . Ð% ß !Ñ ? 8Ä∞ 8Ä∞
75 4. Closed Sets and Operators on Sets
Definition 4.1 Suppose Ð\ß .Ñ is a pseudometric space and that J©\. We say that J is closed in Ð\ß .Ñ if \ J is open in Ð\ß .Ñ .
From the definitions: JÐ\ß.Ñ is closed in iff \J is open in Ð\ß.Ñ iff for all B−\J , there is an % 0 for which FÐBÑ©\J% iff for all B−\J , there is an % 0 for which F% (x ) ∩J œ g .
The closed sets are completely determined by the open sets and vice-versa so that in any space Ð\ , .Ñ the collection of closed sets and the collection of open sets, g. , contain exactly “the same information.”
The close connection between the closed sets and the open sets is illustrated in the following theorem.
Theorem 4.2 For any pseudometric spaceÐ\ß .Ñ ,
i) g\ and are closed ii) if is closed for each then is closed J−EßJααα α −E iii) if are closed, then 8 is closed. J"8 ß ÞÞÞß J3œ3 J 3 (Conditions ii) and iii) say that the collection of closed sets is closed under intersections and finite unions.)
Proof If we take complements, then these statements follow from the corresponding properties of open sets. Since g\ and are open, the complements \gœ\\\œg and are closed.
Suppose J−E\J−Eαα is closed for each αα . Then is open for each , so is open, and therefore the complement α−EÐ\Jα Ñ \Ðα−E Ð\Jα ÑÑ is closed. œ\Ð\JÑœJαα−Eαα −E The proof of iii) uses the fact that a finite intersection of open sets is open. ñ
Exercise: Give an example to show that an infinite union of closed sets need not be closed.
Example 4.3
1) In ‘‘ , the interval Ò!ß "Ó is closed since its complement Ò!ß "Ó œ Ð ∞ß !Ñ ∪ Ð"ß ∞Ñ is open. Equivalently, we can say that Ò!ß "Ó is closed because for each B  Ò!ß "Óß there is an %%% ! for which F% ÐBÑ ∩ Ò!ß "Ó œ ÐB ß B Ñ ∩ Ò!ß "Ó œ gÞ
2) A set can be neither open nor closed: for example, consider the following subset of ‘ : Ò!ß "Ñ, and .
76 3) A set can be both open and closed these terms are not mutually exclusive . Such sets in Ð\ß .Ñ are called clopen sets. For example, g and \ are clopen in every pseudometric space Ð\ß .Ñ . Sometimes there are other clopen sets and sometimes not. In the space \ œ Ò!ß "Ó ∪ Ò$ß %Ó with the usual metric . , the set Ò!ß "Ó is clopen. But in ‘ , for example, gÞ and ‘ are the only clopen subsets (This fact is not too hard to prove but it is also not obviousÐœÑ the proof depends on the completeness property “least upper bound property” in ‘. We will prove this fact later when we need it in Chapter V.)
4) In any pseudometric space Ð\ß .Ñ , the set ÖB − \ À .Ð+ß BÑ Ÿ% × œ J is a closed set.
To see this, suppose CÂJ , Then .Ð+ßCÑœ$% Þ Let %" œ $% ! . Then F%" ÐCÑ∩J œ gÞ
(If D − F%" ÐCÑ ∩ J , then we would have .Ð+ß CÑ Ÿ .Ð+ß DÑ .ÐDß CÑ %% " œ % Ð $% Ñ œ $ , which is false.)
ÖB − \ À .Ð+ß BÑ Ÿ% × is called the closed ball centered at B with radius %.
""" " For example, Ò!ß"ÓœÖB−‘ À.Ð### ßBÑœlB lŸ × is the “closed ball” centered at # Þ
5) Let \ œ Ò!ß "Ó ∪ Ò#ß &Ñ with the usual metric . .
Ò!ß "Ó and Ò#ß &Ñ are both clopen in Ð\ß .Ñ . Ò$ß %Ñ is neither open nor closed in Ð\ß .ÑÞ
Notice again that “open” and “closed” are not absolute terms: whether a set E is open (or closed) depends on what larger space \E that “lives in.”
6) Let . be the discrete unit metric, so that g. is the discrete topology : every subset of \ is open. Then every subset of \ is clopen.
7) Let .\ÞFÐBÑœ\B−\! be the trivial pseudometric on Then % for every and every % . Therefore a union of balls must be either \g or (in the case of the union of an empty collection of balls), so g. œ Ögß \× , which is called the trivial topology on \ . Since g and \ must be open in any space Ð\ß .Ñ , the trivial topology is the smallest possible topology on \ . In Ð\ß .Ñß the only closed sets are g and \ .
Using the open and closed sets in Ð\ß .Ñ , we can define some useful “operators” on subsets of \Þ An “operator” creates a new subset of \ from an old one.
Definition 4.4 Suppose Ð\ß .Ñ is a pseudometric space and E © \Þ
The interior of in int is open and E \ œœÖ\ ESÀS S ©E× The closure of in X cl is closed and } E œÖJÀJJ\Eœ ªE The frontier (or boundary) of E in \ œEœ∩ Fr\\ clE\EÑ cl(\
We will omit the subscript “\\ ” when the context makes clear the _space in which the operations are being performed. Sometimes intEE and cl are denoted EE ° and respectively. Some books use the
77 notation `E for Fr E , but the symbol `E has a different meaning in algebraic topology so we will avoid using it here.
Theorem 4.5 Suppose Ð\ß .Ñ is a pseudometric space and that E © \Þ Then
1) a) int EE©E is the largest open subset of (that is, if OO is open and , then O ©Eint ). b) E© is open iff Eœ int E (since int E E , the equality is equivalent to E© int EÑ . c) B− int E iff there is an open set S such that B−S©E iffb % 0 such that FÐBÑ©E% . Informally, we can think of 1c) as saying that the interior of E consist of those points “comfortably inside E surrounded by a small cushion,” that is, points not “on the edge of E .”
2) a) cl EEE is the smallest closed set containing (that is, if JJª is closed and , 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 S containing B , S∩EÁg iff for every % ! , FÐBÑ∩EÁg% . Informally, 2c) states that -6E consists of the points in \ that can be approximated arbitrarily closely by points from within the set EÞ
3) a) FrEEœÐ\EÑ is closed and Fr Fr . bÑE is clopen iff Fr Eœg . c)B− Fr E iff for every open set S containing B , S∩EÁg and S∩Ð\EÑÁg iff for every % ! FÐBÑ∩E%% Ág and FÐBÑ∩Ð\ EÑÁg . Informally, 3c) states that FrE\ consists of those points in that can be approximated arbitrarily closely both by points from within E and by points from outside E.
Proof
1) a) Clearly intE©E . If S open and S©E , then S is one of the sets whose union is int,ES©E so int.
b) intEEœEE is open (it is a union of open sets), so if int , then is open. Conversely, if EE is open, then is the largest open subset of EEœEÞ , so int
c) Since intEB−EB−S©E is a union of open sets, it is clear that int iff for some open set SÞ Since a open set is open iff it is a union of % -balls, the remainder of the assertion is obviously true.
2) Exercise
3) a) FrE is closed because it is an intersection of two closed sets, and FrÐ\ EÑ œ cl Ð\ EÑ ∩ cl Ð\ Ð\ EÑÑ œ cl Ð\ EÑ ∩ cl ÐEÑ œ Fr EÞ
78 b) If E is clopen, then so is \ E . Therefore Fr ÐEÑ œ cl ÐEÑ ∩ cl Ð\ EÑ œ E ∩ Ð\ EÑ œ gÞ Conversely, if cl ÐEÑ ∩ cl Ð\ EÑ œ g , then clE©\ cl Ð\EÑ©\Ð\EÑœE , so E is closed. Similarly we show that \E is closed, so E is clopen.
c) B− Fr E iff B in both cl E and cl Ð\EÑ . By 2c), this is true iff each open set containing BE\EÞ intersects both and Since an open set is a union of % -balls, the remainder of the assertion is clearly true. ñ
Notice that section c) in each part of Theorem 4.5, there is a criterion that lets you decide whether B is in one of the sets intEE , cl , or Fr E by using only the open sets ß and not mentioning % -balls. This is important! It means that if we change the metric ..EE to an equivalent metric w , then int , cl , and Fr E.. do not change, since and w produce the same open sets. In other words, we can say that int, cl, and Fr are topological operators: they depend on only the topology, and not on the particular metric that produced the topology. For example if E©‘8 , then E will have the same interior, same closure, and same frontier whether we measure distances using the usual metric.. , the taxicab metric> , or the max-metric.‡ .
Example 4.6 ()Be sure you understand each statement!
1) In ‘‘‘‘‘‘ : int œœ cl Fr œg int ‘‘œg cl œ Fr œ
intÒ!ß "Ñ œ Ð!ß "Ñ cl Ò!ß "Ñ œ Ò!ß "Ó Fr Ò!ß "Ñ œ Ö!ß "× FrFr Б Ñ œ Fr Ð Ñ œ gÞ
In any spaceÐ\ß .Ñ , it is obviously true that int Ð int EÑ œ int E and cl Ð cl EÑ œ cl E . But this need not be true for Fr, as the last example shows. ( It is true that FrÐÐ Fr Fr EÑќРFr Fr EÑ in any space \Þ But this is not a useful fact, and it is also not interesting to prove.)
2) \œÒ!ß#Ñ (with the usual metric)
cl\\Ò!ß "Ñ œ Ò!ß "Ó int Ò!ß "Ñ œ Ò!ß "Ñ cl\\Ò"ß #Ñ œ Ò"ß #Ñ int Ò"ß #Ñ œ Ð"ß #Ñ
Fr\Ò!ß "Ñ œ Ò!ß "Ó ∩ Ò"ß #Ñ œ Ö"× !ÂFr\ Ò!ß"Ñ because ! cannot be “approximated arbitrarily closely” by points from \Ò!ß"Ñ.
3) Suppose .\E©\E is the discrete unit metric on . If , then is clopen so we get clEœE , int EœE , and Fr EœgÞ Suppose .\E is the trivial pseudometric on . If is any nonempty, proper subset of \ , then cl Eœ\ , int Eœg , and Fr Eœ\Þ
4) In Ðj# ß .Ñß let E be the set of sequences with all terms rational:
E œ ÖB œ ÐB3# Ñ − j À a3ß B 3 − × œ ∩ j # Þ
79 We claim that clEœj#3# in other words, that any point CœÐCÑ−j can be approximated arbitrarily closely by a point from E! . So let % . We must show that FÐCÑ∩EÁgÞ%
∞∞ ## %# Since CRC33 cinverges, we can pick an such that # . Using this value of 3œ"3œR" % R, choose rational numbers +333 so that l+ C l for 3 œ "ß ÞÞÞß RÞ #R Define + œ Ð+"R ß ÞÞÞß + ß !ß !ß !ß ÞÞÞ ÑÞ Then + − E and
∞R∞ ### .Ð+ßCÑœ Ð+CÑœ33 Ð+CÑ 33 Ð+CÑ 33 3œ" 3œ" 3œR"
R∞ œÐ+CÑC# # R†œ%%## %%# œÞ 33 3 #R # 3œ" 3œR"
Therefore +−FÐCÑ∩EÞ%
We also claim that intEœgÞ To prove this, we need to show that if BœÐBÑ−Eß3 then no ball centered at BE is a subset of . To see this, pick any % ! and choose an irrational ClCBlÞB""" such that % Modify by changing B " : define C œ ÐC"# ß B ß ÞÞÞß B 8 ß ÞÞÞÑÞ Then .ÐBß CÑ œ lB " C " l % and C  Eß so FÐBÑ©% ÎE .
What is FrE ?
5) In any pseudometric space Ð\ß .Ñ , F% Ð+Ñ is a subset of the closed set ÖB − \ À .Ð+ß BÑ Ÿ%% × . Therefore cl F% Ð+Ñ © ÖB − \ À .Ð+ß BÑ Ÿ ×Þ But these two sets are not necessarily equal : sometimes the closed ball is larger than the closure of the open ball FÐ+Ñ% ! For example, suppose . is the discrete unit metric on a set \l\l"Þ where Then Ö+× œ F"" Ð+Ñ œ cl F Ð+Ñ § \ œ ÖB − \ À .Ð+ß BÑ Ÿ "× Á
Definition 4.7 Let Ð\ß .Ñ be a pseudometric space and suppose that H © \Þ We say that H is dense in Ð\ß .Ñ if cl H œX . The space Ð\ß .Ñ is called separable if it possible to find a countable dense set H in X . ( Note the spelling: “sepa rable,” not “sepe rable.” )
Since “separable” is defined in terms of the closure and the closure operator depends only on the topology, not the particular metric that generates the topology. Therefore separability is a topological property: if Ð\ß .Ñ is separable and .ww µ . , then Ð\ß . Ñ is also separable.
More informally, “H\ is dense in ” means that each point B−\ can be approximated arbitrarily closely by a point from H\ . A countable dense set in (if one exists) is a “small” set which can be used to approximate any point in \ arbitrarily closely.
80 Example 4.8
1)‘88 is separable because is a countable dense set in ‘ 8à in particular, is a countable dense set in ‘‘ , so is separable. is an example of an uncountable dense subset of ‘ .
2) Any countable space Ð\ß .Ñ is separable, because \ is dense in \ .
3) Supposeg. is the discrete topology on \Ð\ß.Ñ\ . Then is separable iff is countable (because any proper subset of \ is closed and therefore not dense). If g. is the trivial topology and \Ág , then every nonempty subset H is dense (why? ). Therefore Ð\ß .Ñ is separable because, for example, each singleton set ÖB× is dense.
4) The set Eœ ∩j## is dense in j (see Example 4.6(4) ). This set E is an " uncountable dense set because every sequence of rationals ÐB33 Ñ with lB l Ÿ3 is in E (why? ), and there are -Ðjß.ÑH such sequences (why? ). However, # is separable. Can you find a countable dense set ? ÐÑThe computation in Example 4.6.4 might give you an idea.
Definition 4.9 For nonempty subsets Eß F in a pseudometric space Ð\ß .Ñ , we define the distance between them by distÐEß FÑ œ inf Ö.Ð+ß ,Ñ À + − E and , − F×Þ
Although it is an abuse of notation “. ” we usually abbreviate dist ÐEß FÑ by .ÐEß FÑÞ Going one step further, we also write.Ð+ß FÑ as an abbreviation for .ÐÖ+×ß FÑÞ
Of course, if +−F , then .Ð+ßFÑœ!Þ But the converse may not be true. For example, let "" # F œ Ö"ß#8 ß ÞÞÞß ß ÞÞÞ× ©‘‘ Þ Then .Ð!ß FÑ œ ! even though !  FÞ Similarly, if E is the C -axis in # " and F œ ÖÐBß CÑ −‘ À C œB × , then E and F are disjoint, closed sets but .ÐEß FÑ œ !Þ
The “distance from a point to a set” can be used to describe the closure of a set.
Theorem 4.10 Suppose E©\ , where Ð\ß.Ñ is a pseudometric space. Then B− cl E iff .ÐBß EÑ œ !Þ
Proof B− cl E iff for every % ! , FÐBÑ∩EÁg% iff for every %%! there is a C−E with .ÐBßCÑ iff .ÐBß EÑ œ !Þ ñ
81 Exercises
E10. Let Ð\ß .Ñ be a pseudometric space. Prove or disprove each statement:
a) FÐBÑ% is never a closed set. b) if E©\ , then Fr Eœ cl E int EÞ c) for any E©\ , diam ÐEÑœ diam (cl EÑÞ ÐThe diameter of a set A in a metric space is defined to be sup Ö.ÐBß CÑ À Bß C − E× Ÿ ∞ .Ñ d) for any E©\ , diam ÐEÑœ diam (int EÑÞ e) for any E©\ , int ÐE∪FÑœ int E∪ int FÞ f) for every B − \ and %% !ß cl ÐF% ÐBÑÑ = ÖC− \ À .ÐBß CÑ Ÿ ×Þ
E11. a) Give an example of a metric space Ð\ß .Ñ with a proper nonempty clopen subset. b) Give an example of a metric space Ð\ß .Ñ and a subset that is neither open nor closed. c) Give an example of a metric space Ð\ß .Ñ and a subset E for which every point in E is a limit point of E . ( Note: a point BES is called a limit point of a set if, for every open set containing B, S ∩ ÐE ÖB×Ñ Á g .) d) Give an example of a metric space Ð\ß .Ñ and a nonempty subset A such that every point is a limit point of AAX but intÐEÑ œ g . Can you also arrange that is closed in ? e) For each of the following subsets of ‘ , find the interior, closure and frontier (“boundary”) in ‘‘ . Which points of the set are isolated in ? which points of the set are isolated in the set ?
i) EœÖ781 À7ß8− × "" ii) Fœ Ö78 À7ß8−™ ×
E12. An infinite union of closed sets need not be closed. However if infinitely many closed sets are “spread out enough” from each other, their union is closed. Parts a) and b) illustrate this.
a) Suppose that for each 8−‘ ßJ88 is a closed set in and that J ©Ð8ß8"Ñ . Prove that ∞ is closed in . 8œ"J8 ‘
b) More generally: suppose for α −E , each Jα is a closed set in Ð\ß.Ñ and that for each point B−\ there is an % ! such that FÐBÑ% has nonempty intersection with at most finitely many 's. Prove that is closed in . (Notice that b a. Why? ) JJÐ\ß.Ñααα−E Ê
E13. a) Give an example of #Þ- subsets of ‘‘ all of which have the same closure Do the same in # .
b) Prove or disprove: there exist #-# subsets of ‘ such that any two have different closures. Can you do the same in ‘ ?
1 E14. The Hilbert cube , HH , is a certain subset of jLœÖB−jÀlBlŸ##3 : i ×Þ Prove that is closed in j# . Is H also open? (Prove or disprove)
82 E15. A subset E\KE of a space is called a $ set if can be written as a countable intersection of open sets; EJ is called an 5 set if it can be written as a countable union of closed sets.
Note: Open sets are often denoted letters like SY , , or Z (from o pen, and from the French ouvert), and sometimes by the letter K — from older literature where the German word is “Gebiet”. Closed sets often are denoted by the letter F — from the French “ferme.” Of course using these letters is just a common tradition but many topologists follow it and would usually wince to read something like “let J be an open set.” The names KJ$5 and go back to the classic book Mengenlehre of the German mathematician Felix Hausdorff. The 5$ and the in the notation represent abbreviations for the German words used for union and intersection: Summe and Durchschnitt.
a) Prove that in a pseudometric space (\ß.Ñ every closed set is a K$ set and every open set is an J5 set.
b) Find the error in the following argument which “proves” that every subset of ‘ is a G$ set:
Let A©B−œÖÐBÑÀB−Eב . For A, let J88 B" . J is open for each n − . 8 ∞∞ Since ÖB× œ B" ÐBÑ , it follows that A œ J8 , so A is a countable 8œ"8 8œ" intersection of open sets, that is, A is a G$ set.
c) In the argument in part b), the truth is that ∞ J ? 8œ" 8 œ
d) Suppose we enumerate the members of : B"# ß B ß ÞÞÞß B 8 ß ÞÞÞ . For each 8 , consider the "" ∞ interval F" ÐB88 Ñ œ ÐB ß B 8 Ñ ©‘‘ and let N 8 œ F" ÐB 88 ÑÞ Is N œ ? 8 88 8œ" 8
E16. Let Ð\ß .Ñ be a pseudometric space. Suppose that for every % ! , there exists a countable subset DX% of with the following property: aB − \ß bC − H% such that .ÐBß CÑ % . Prove that Ð\ß .Ñ is separable.
E17. Suppose that \.\ is an uncountable set and is any metric on which produces the discrete topology. (Such a metric d does not have to be a constant multiple of the discrete unit metric: see Example 2.14.2). Show that for some % ! there is an uncountable subset E of \ such that.ÐBß CÑ % for all B Á C − E
E18. Let Ð\ß .Ñ be an infinite metric space. Prove that there exists an open set Y such that both Y and \Y are infinite. (Hint: Consider a non-isolated point, if one exists.)
E19. A metric space Ð\ß .Ñ is called extremally disconnected if the closure of every open set is open. ( Note: “extremally” is the correct spelling; the word is not the same as the everyday word “extremely.” ) Prove that if Ð\ß .Ñ is extremally disconnected, then the topology g. is the discrete topology.
83 E20. Since “dist” provides a measure of “distance” between nonempty subsets of Ð\ß .Ñ , one might ask whether Ðc Ð\Ñ Ög×ß dist Ñ is a metric (or pseudometric) space. Is it?
E21. Suppose ÐB8!8 Ñ is a sequence in Ð\ß .Ñ . We say that B is a cluster point of ÐB Ñ if for every open set SB containing !5 and for all 8−ßb58B−S such that . (This is clearly equivalent to saying that a!% and a8−ßb58 such that B−FÐBÑ5!% .) Informally, B! is a cluster point of (BB8 ) if the sequence is “frequently in every open set containing ! .”
a) Show that there is a sequence in ‘‘ for which every <− is a cluster point.
b) A neurotic mathematician is walking along ‘ from 0 toward 1. Halfway to 1, she (or he) remembers that she forgot something at 0 and starts back. Halfway back to 0, she decides to go to 1 anyway and turns around, only to change her mind again after traveling half the remaining distance to " . She continues in this back-and-forth fashion forever. Find the cluster th point(s) of the sequence ÐB88 Ñ , where B is the point where she reverses direction for the 8 time.
84 5. Continuity
Suppose +−E©‘‘ and that 0ÀEÄ Þ In elementary calculus, the set E is usually and interval, and the idea of continuity at a point +E in is introduced very informally. Roughly, it means that “if B is a point near + in the domain, then 0ÐBÑ is near 0Ð+ÑÞ ” In advanced calculus or analysis, the idea of “continuity of 0+ at ” is defined carefully. The intuitive version of continuity stated in terms of “nearness” is made precise by measuring distances:
0+a!b! is continuous at means: %$ such that
if B − E and lB +l $%, then l0ÐBÑ 0Ð+Ñl Þ
We say that “00 is continuous” if is continuous at every point +E in its domain .
An important thing to notice is that the definition is made using the distance function in ‘ : .ÐBß +Ñ œ lB +l and .Ð0ÐBÑß 0Ð+ÑÑ œ l0ÐBÑ 0Ð+Ñl . Since we have a way to measure distances in pseudometric spaces, we can make an a completely similar definition of continuity for functions from one pseudometric space to another.
Definition 5.1 S uppose 0À\Ä] where Ð\ß.Ñ and Ð]ß=Ñ are pseudometric spaces Þ If +−\ and 0À\Ä], we say 0 is continuous at a if a%$ ! b ! such that: if .Ð+ßBÑ $ , then =Ð0ÐBÑß 0Ð+ÑÑ % Þ
Notice that the sets \] and may have two completely unrelated metrics .=. and : measures distances in domÐ0Ñ and = measures distances in ] . But the idea is exactly the same as in calculus: “continuity of 0+ at ” means, roughly, that “points B near + in the domain \ have images 0ÐBÑ near 0Ð+Ñ in ] .”
Theorem 5.2 S uppose Ð\ß .Ñ and Ð] ß =Ñ are pseudometric spaces Þ If + − \ and 0 À \ Ä ] , then the following statements are equivalent:
1) 0 is continuous at a 2) a%$ ! b ! such that 0 ÒF$% Ð+ ÑÓ © F Ð0Ð+ÑÑ " 3) a%$ ! b ! F$% Ð+Ñ © 0 ÒF Ð0Ð+ÑÑÓ 4) aR © ] : if 0()aaN − int R , then −int 0 "[ ].
Proof It is clear that conditions 1) -# ) - 3) are just equivalent restatements the definition of continuity at + phrased in terms of images and inverse images of balls. Condition 4), however, seems a bit strange. We will show that 3) and 4) are equivalent.
3)Ê0Ð+Ñ−RÞ 4) Suppose int By definition of interior, there is an % ! such that " " F%% Ð0Ð+ÑÑ © R, so that 0 ÒF Ð0Ð+ÑÑÓ © 0 ÒRÓÞ By 3), we can pick $ ! so that " " " + − F$% Ð+Ñ © 0 ÒF Ð0Ð+ÑÑÓ © 0 ÒRÓ. Since the $ -ball at + is an open subset of 0 ÒRÓ , we get that +−int 0" ÒRÓ , as desired.
4)Ê 3) Suppose % ! is given. Let R œ F% Ð0Ð+ÑÑÞ Then R is open and 0Ð+Ñ − R " " œint RÞ We conclude from 4) that + − int 0 ÒRÓ œ int 0 ÒF% Ð0Ð+ÑÑÓÞ Therefore for some $ !ß " " F$% Ð+Ñ ©int 0 ÒF Ð0Ð+ÑÑÓ © 0 ÒF % Ð0Ð+ÑÑÓ , so 3) holds. ñ
85 Definition 5.3 If R © \ and B − int Rß then R is called a neighborhood of B in Ð\ß .ÑÞ Thus, RB is a neighborhood of if there is an open set SB−S©R such that .
Notice that:
1) The term “neighborhood” goes together with a point B−\Þ We might say “ S is an open set in \R\R ,” but we would never say “ is a neighborhood in ” but rather “ is a neighborhood of B in \.”
2) A neighborhood R of B need not be an open set . Be aware, however, that in some texts “a neighborhood of BB ” means “an open set containing .” It's not really important which way one makes the definition of neighborhood (each version has its own technical advantages), but it is important that we all agree in these notes. So, in ‘8 for example, we say that the closed ball JœÖÐ+ßBÑÀ .Ð+ß BÑ Ÿ% × is a neighborhood of + ; in fact the closed ball is a neighborhood of any point B in its interior. A point B.Ð+ßBÑœJJ for which % is in , but is not a neighborhood of such a point B .
The following observation is almost trivial but it is important enough to state and remember.
Theorem 5.4 A subset RÐ\ß.Ñ in is open iff R is a neighborhood of each of its points.
Proof Suppose RB−RÞB−RœRÞRR is open and Then int Since int is open, is a neighborhood of B . Conversely, if RB−Rß is a neighborhood of each of its points, then for every we have B−int RÞ Therefore R© int R , so Rœ int R and R is open. ñ
With this new terminology, we can restate the equivalence of 1) and 4) in Theorem 5.2 as:
0+ is continuous at iff whenever R0Ð+ÑÐÑ0ÒRÓ+ÐÑ is a neighborhood of in ]\ , then " is a neighborhood of in .
This tells us something very important. Interiors (and therefore neighborhoods of points) are defined in terms of the open sets (without needing to mention the distance function), so the neighborhoods of a point B depend only on the topology, not on the specific metric that generates the topology. Therefore whether or not 0 is continuous at + does not actually depend on the specific metrics but only on the topologies in the domain and range. In other words, “continuity at + ” is a topological property.
For example, the function sinÀÄ‘‘ is continuous at each point + in ‘ , and this is remains true if we measure distances in the domain with, say, the taxicab metric .> and distance in the range with the max-metric.‡ since these are both equivalent to the usual metric . on ‘ .
We now define “0 is a continuous function” in the usual way.
Definition 5.5 Suppose Ð\ß .Ñ and Ð] ß =Ñ are pseudometric spaces. We say that 0 À \ Ä ] is continuous if 0\ is continuous at each point of .
Theorem 5.6 Suppose 0À\Ä] where Ð\ß.Ñ and Ð]ß=Ñ are pseudometric spaces. The following are equivalent:
86 1) 0 is continuous 2) if S]0ÒSÓ\ is open in , then " is open in 3) if J]Ò is closed in , then 0JÓ" is closed in \ .
Proof 1)ÊS]B−0ÒSÓS 2) Suppose is open in and that " . Since is a neighborhood of 0ÐBÑ and 0B is continuous at , we know from Theorem 5.2 that 0ÒSÓB" is a neighborhood of . Therefore 0ÒSÓ" is a neighborhood of each of its points, so 0ÒSÓ " is open.
2)Ê 3) If J is closed in ] , then ] J is open. By 2), 0" Ò]JÓ œ\0 " ÒJÓ is open in \ß so \ Ð\ 0" ÒJ ÓÑ œ 0 " ÒJ Ó is closed in \ .
3)Ê 2) Exercise
2)Ê+−\R 1) Suppose and that is a neighborhood of 0Ð+Ñ] in , so that 0Ð+Ñ−int R©RÞ By 2), 0" Ò int RÓ is open in \ , and +−0 " Ò int RÓ©0 " ÒRÓ . Therefore 0ÒRÓ" is a neighborhood of +Þ Therefore 0 is continuous at + . Since + was an arbitrary point in \0 , is continuous. ñ
Notice again that Theorem 5.6 shows that continuity is completely described in terms of the open sets (or equivalently, the closed sets), and the proof of the theorem is phrased entirely in terms of open (closed) sets, without any explicit mention of the pseudometrics on \] and . Replacing .= and with equivalent pseudometrics would not affect the continuity of 0 .
Theorem 5.7 Suppose Ð\ß .Ñ , Ð] ß =Ñ and Ð^ß >Ñ are pseudometric spaces and that 0 À \ Ä ] and 1À] Ä^Þ If 0 is continuous at ag−\ and is continuous at 0 (a ) −] , then 1‰0 is continuous at a. (Therefore, if 01 and are continuous, so is 1‰0Þ )
Proof If R is a neighborhood of 1Ð0Ð+ÑÑ , then 1" ÒRÓ is a neighborhood of 0Ð+Ñ , because 1 is continuous at 0Ð+Ñ . Since 0 is continuous at + , 0" Ò1 " ÒRÓÓ is a neighborhood of + . But 0" Ò1 " ÒRÓÓœ Ð1‰0Ñ " ÒRÒßso 1‰0 is continuous at + . ñ
Example 5.8 "ßif B œ ! 1) Suppose 0À‘‘ Ä is given by 0ÐBÑœ Þ Then Rœ ("$ ß Ñ is a !ßif B Á ! ## neighborhood of 0Ð!Ñ œ "ß but 0" ÒRÓ œ Ö!× is not a neighborhood of !Þ Therefore 0 is not continuous at 0. To see the same thing using slightly different language: " 0! is not continuous at because, choosing %$ œß!# there is no choice of such that 0ÒF$% Ð!ÑÓ © F Ð0Ð!ÑÑ œ F % Ð"ÑÞ
87 2) If 0 À Ð\ß .Ñ Ä Ð] ß =Ñ is a constant function, then 0 is continuous. To see this, g-ÂSif suppose 0ÐBÑ œ - for every B . If S is open in ] , then 0" ÒSÓ œ . \-−Sif In both cases, 0ÒSÓ" is open.
"B−if 3) Suppose 0À‘‘ Ä is given by 0ÐBÑœ . Then 0 is not continuous !B−if at any point +−‘‘ (why?). However 0l œ1À Ä is continuous at every point of , because 1 is a constant function.
There is a curious old result called Blumberg's Theorem which states: For any0À‘‘ Ä , there exists a dense set H© ‘ such that 0lHœ1ÀHÄ‘ is continuous. Blumberg's Theorem is rather difficult to prove, and not very useful.
4) Suppose 0ß1 À \ Ä‘ where Ð\ß.Ñ is a pseudometric space. Since these functions are real-valued, it makes sense to define functions 01ß01ß0†1 and 0 1 in the obvious way. For example, Ð0 1ÑÐBÑ œ 0ÐBÑ 1ÐBÑ , where the “ ” on the right is ordinary addition in ‘ . If 0 and 1 are continuous at a point +−\ß then the functions 01ß01ß and 0 0†1 are also continuous at + ; and 1 is continuous at + if 1Ð+ÑÁ 0. The proofs are just like those given in calculus Ð\œÑ where ‘ For example, consider 01 : given % ! , then (because 0ß1 are continuous at +ß ) we can find $$"#! and ! so that
% if .ÐBß +Ñ $" ßthen l0ÐBÑ 0Ð+Ñl # and % if .ÐBß +Ñ $# ßthen l1ÐBÑ 1Ð+Ñl #
Let $$$œÖß×Þ.ÐBß+Ñ min "# Then if $ , we have
lÐ0 1ÑÐBÑ Ð0 1ÑÐ+Ñl œ l0ÐBÑ 0Ð+Ñ 1ÐBÑ 1Ð+Ñl %% Ÿ l0ÐBÑ 0Ð+Ñ || 1ÐBÑ 1Ð+Ñl ## œ %
You can find at the other proofs in an analysis textbook.
5) For B œ ÐB"# ß B ß ÞÞÞß B 8 ß ÞÞÞÑ − j # , define 0 À j # Ä‘ by 0ÐBÑ œ B " Þ ( 0 is a “projection” of jÞ# onto ‘ ) Then 0B is continuous at . To see this, suppose % ! , and let $%œ Þ If C œ ÐC"# ß C ß ÞÞÞß C 8 ß ÞÞÞÑ − F$ ÐBÑ , then ∞ # "Î# l0ÐBÑ 0ÐCÑl œ lB"" C l Ÿ .ÐBß CÑ œ Ð ÐB 33 C Ñ Ñ $% œ , 3œ" so 0ÒF$% ÐBÑÓ © F Ð0ÐBÑÑ .
A similar argument shows that the projection 1ÐBÑ œ B8 is also continuous, and an argument only slightly more complicated would show, for example, that the $ projection function 2Àj#$*"" Ä‘ given by 2ÐBÑœÐBßBßB Ñ is continuous.
88 6) If + is an isolated point in Ð\ß .Ñ , then every function 0 À Ð\ß .Ñ Ä Ð] ß =Ñ is continuous at +Þ To see this, suppose R is a neighborhood of 0Ð+Ñ . Then + − Ö+× © 0" ÒRÓ . But Ö+× is open in \ , so 0 " ÒRÓ is a neighborhood of + . If g. happens to be the discrete topology, then every point in \ is isolated so 0 is continuous. (In this case, we could argue instead that whenever S] is open in then 0ÒSÓ" must be open in \ because every subset of \ is open.)
7) A function 0 À Ð\ß .Ñ Ä Ð] ß =Ñ is called an isometry of \ into ] if it preserves distances, that is, if .Ð+ß ,Ñ œ =Ð0Ð+Ñß 0Ð,ÑÑ for all +ß , − \Þ An isometry is clearly continuous (given %$%! , choose œ ). Note that if .0 is a metric , then one-to-one (Why? ). If 0 happens to be a bijection, we say that Ð\ß .Ñ and Ð] ß =Ñ are isometric to each other. In that case, it is clear that the inverse function 00" is also an isometry, so " isalso continuous.
Theorem 5.9 If Ð\ß .Ñ is a metric space and B Á C − \ , then there exist open sets Y and Z such that B−Y, C−Z and Y∩Z œgÞ ( More informally: distinct points in a metric space can be separated by disjoint open sets.)
Proof Since B Á Cß .ÐBß CÑ œ$ !Þ Let Y œ F$$ ÐBÑ and Z œ F ÐCÑÞ These sets are open, and if ## there were a point D − Y ∩ Z , we would have a contradiction: .ÐBß CÑ Ÿ .ÐBß DÑ .ÐDß CÑ $$ œ##$. ñ
Theorem 5.9 may not be true if..\ is not a metric. For example, if is the trivial pseudometric on , then the only open sets containing BCYœZœ\Þ and are )
Example 5.10
1) Suppose 0 À Ð\ß .Ñ Ä Ð] ß =Ñ , where . is the trivial pseudometric on \ and ] is any metric space. We already know that if 00 is constant, then is continuous. If 0 is not constant, then there are points +ß , − \ for which 0Ð+Ñ Á 0Ð,ÑÞ Since =YZ] is a metric, we can pick disjoint open sets and in with g+−0ÒYÓ()" 0Ð+Ñ − Y and 0Ð,Ñ − Z Þ Then 0" ÒYÓ Á \,Â0ÒYÓ()" " Since g. œ Ögß \× , we see that 0 ÒY Ó is not open so 0 is not continuous. So, in this situation: 00 is continuous iff is constant.
2) Suppose .= is the usual metric and is the discrete unit metric on ‘ . Let 3ÀБ‘ ß.ÑÄÐ ß=Ñ is the identity map 3ÐBÑœBÞ For every open set S in Ð ‘ ß.Ñß the image set 3ÒSÓ is open in Б ß =Ñ , but this is not the criterion for continuity: in fact, 0 is not continuous at any point . The criterion for continuity is that the inverse image of every open set must be open.
Example 5.10.2 leads us to a definition.
89 Definition 5.11 A function 0 À Ð\ß .Ñ Ä Ð] ß =Ñ is called an open function or open mapping if: whenever O is open in \ß then the image set 0 ÒSÓ is open in ] . Similarly, we say 0 a closed mapping if whenever J\ is closed in , then the image set 0ÒJÓ is closed in ] .
The identity mapping 33 in Example 5.10.2 is both open and closed but is not continuous. You can # convince yourself fairly easily that the projection function 1‘BBÀÄ ‘ given by 1 ÐBßCÑœB is open " # and continuous, but it is not closedJœÖÐBßCÑÀCœ× for example, the set B is a closed set in ‘ but 1‘BÒJ Ó œ Ð!ß ∞Ñ is not closed in . The general observation is that for a function 0 À Ð\ß .Ñ Ä Ð] ß =Ñß the properties “open”, “closed”, and “continuous” are completely independent. You should provide other examples: for instance, a function that is continuous but not open or closed.
With just the basic ideas about continuous functions, we can already prove some rather interesting results.
Theorem 5.12 Suppose 0ß 1 À Ð\ß .Ñ Ä Ð] ß =Ñ , where . is a pseudometric and = is a metric . Let H be a dense subset of \Þ If 0 and 1 are continuous and 0lH œ 1lH, then 0 œ 1 . (More informally: if two continuous functions with values in a metric space agree on a dense set, then they agree everywhere.)
Proof Suppose 0Á1Þ Then 0Ð+ÑÁ1Ð+Ñ for some point +−\Þ Since = is a metric, we can find disjoint open sets YZ]0Ð+Ñ−Y1Ð+Ñ−ZÞ01 and in with and Since and are continuous at + , there are open sets Y"" and Z in \ which contain + and satisfy 0ÒYÓ©Y " and 1ÒZÓ©Z " . We know +− cl H . Since Y"" ∩Z is an open set containing +ß there must be a point . − ÐY"" ∩ Z Ñ ∩ H. Then 0Ð.Ñ Á 1Ð.Ñ because 0Ð.Ñ − Y ß 1Ð.Ñ − Z and Y ∩ Z œ gÞ Therefore 0lH Á 1lHÞ ñ
Example 5.13 If 0ß1 − GБ Ñ and 0l œ 1l , then 0 œ 1 by Theorem 5.12. In other words, the mapping F‘ÀGÐÑÄGÐÑ given by F Ð0Ñœ0l−GÐÑ is one-to-one. Therefore lGБ Ñl Ÿ lGÐ Ñl Ÿ -i! œ -. On the other hand, each constant function 0ÐBÑ œ < is in GБ‘ Ñ , so lGÐ Ñl -Þ It follows that lGБ Ñl œ - . In other words, there are exactly - continuous functions from ‘ to ‘ .
Example 5.14 Find all continuous functions 0À‘‘ Ä satisfying the functional equation
0ÐBCÑ œ 0ÐBÑ0ÐCÑ for all BßC −‘ ЇÑ
Simple induction shows that for 8 − ß 0ÐB"8" ÞÞÞ B Ñ œ 0ÐB Ñ ÞÞÞ 0ÐB 8 ÑÞ By Ð‡Ñ , 0Ð!Ñ œ 0Ð! !Ñ œ 0Ð!Ñ 0Ð!Ñ , so 0Ð!Ñ œ !. Let 0Ð"Ñ œ - . Then 0Ð#Ñ œ 0Ð""Ñ œ 0Ð"Ñ0Ð"Ñ œ - †# 0Ð$Ñ œ 0Ð#"Ñ œ 0Ð#Ñ0Ð"Ñ œ - †#- œ - †$ ã Continuing, we see that 0Ð8Ñ œ -8 for every B − Þ Similarly, for each 7ß8 − ß we have
""""""" 0Ð"Ñ œ 0Ð8888888 ÞÞÞ Ñ œ 0Ð Ñ ÞÞÞ0Ð Ñ œ 80Ð Ñß so 0Ð Ñ œ -Ð Ñ Å 8 terms
90 7" " " "7 0Ð88 Ñ œ 0Ð ÞÞÞ 8 Ñ œ 70Ð 8 Ñ œ 7†-Ð 88 Ñ œ -Ð Ñ Å 7 terms
So far, we have shown that a function 0Ð‡Ñ satisfying must have the formula 0ÐBÑœ-B for every 7 positive rational Bœ8 Þ
77 77 Since !œ0Ð!Ñœ0Ð88 Ð ÑÑœ0Ð 88 Ñ0Ð Ñß we get that
7777 0Ð8888 Ñ œ 0Ð Ñ œ Ð Ñ- œ -Ð ÑÞ
Therefore, 0ÐBÑ œ -B for every B − Þ
So far, we have not used the hypothesis that 01ÀÄ1ÐBÑœ-B is continuous. Let ‘‘ be defined by . Since 0 and 1 are continuous and 0l œ1l , Theorem 5.12 tells us that 0œ1 , that is, 0ÐBÑœ-B for all B−‘.
The continuous functions satisfying the functional equation ‡Ñ were first described by Cauchy in 1821. It turns out that there are also discontinuous functions 0Ð‡Ñ satisfying ; however they are not easy to find. In fact, they must satisfy a nasty condition called “nonmeasurability” (which makes them “extremely discontinuous”).
In calculus, another important is the idea of a convergent sequence (in ‘‘ or 8 ). We can generalize the idea of a convergent sequence in ‘8 to any pseudometric space.
Definition 5.15 A sequence ÐB8 Ñ in Ð\ß .Ñ converges to B − \ if any one of the following (clearly equivalent) conditions holds:
1) a!bR−% such that if nN, then B−FÐBÑ8 % (that is, if the sequence of numbers Ð.ÐB8 ß BÑÑ Ä ! in ‘ )
2) if SB−S©bR− is open and XnN, , then such that if B−S then 8
3) if W, is a neighborhood of BbR− then such that if nN, then B−8 W .
If ÐB88 Ñ converges to \ , we write ÐB Ñ Ä BÞ
Condition= 2) and 3) describe the convergence of sequences in terms of open sets (or neighborhoods) rather than directly using distances. Therefore replacing .. with an equivalent metric w does not affect which sequences converge to which points: sequential convergence is a topological property.
If a sequence ÐB8 Ñ has a certain property T for all 8 some R , we say that “ ÐB8 Ñ eventually has propertyT .” For example, the sequence (0,3,1,7,7,7,7,...) is eventually constant ; the sequence Ð "ß $ß &ß #ß &ß 'ß (ß )ß ÞÞÞß 8ß 8 "ß ÞÞÞÑ is eventually increasing . Using this terminology, we can give a completely precise definition of convergence by saying: ÐB88 Ñ converges to B if ÐB Ñ is eventually in each neighborhood of B .
91 " Example 5.16 1) In ‘ , ÐÑÄ!Þ8
2) Suppose .\ÖB×ÐBÑÄB is the discrete unit metric on . Then each set is open so 8 iff ÐB88 Ñ is eventually in each neighborhood of \ iff ÐB Ñ is eventually in ÖB×Þ In other words, ÐB88 Ñ Ä B iff B œ B eventually Þ Every convergent sequence is eventually constant. At the other extreme, suppose .\ is the trivial pseudometric on . Then every sequence ÐB8 Ñ converges to every point B − \ (since the only neighborhood of B is \ ).
Example 5.16.2 shows that limits of sequences in a pseudometric space do not have to be unique: the same sequence can have many limits. However if .Ð\ß.Ñ is a metric , then sequential limits in must be unique, as the following theorem shows.
Theorem 5.17 A sequence ÐB8 Ñ in a metric space Ð\ß .Ñ has at most one limit.
Proof Suppose BÁC−\ and let YßZ be disjoint open sets with B−Y and C−ZÞ If ÐBÑÄB8 , then ÐB8 Ñ must be eventually in Y Þ Since Y and Z are disjoint, this means that ÐB8 Ñ cannot be eventually in ZÑÐBÑÄ (in fact, the sequence is eventually outside Z , so 8 ÎCÞñ
In a pseudometric space, sequences can be used to describe the closure of a set.
Theorem 5.18 Suppose E©\ , where Ð\ß.Ñ is a pseudometric space. Then B − cl E iff there is a sequence Ð+88 Ñ in E for which Ð+ Ñ Ä B .
Proof First, suppose there is a sequence Ð+88 Ñ in E for which Ð+ Ñ Ä B . If R is any neighborhood of BÐ+Ñ, then 8 is eventually in R . Therefore R∩EÁgB−EÞ , so cl Conversely, suppose B− cl E . Then F" ÐBÑ∩EÁg for each 8− , so we can choose a 8 point +888 − F" ÐBÑ ∩ E . Then Ð+ Ñ Ä B (because .Ð+ ß BÑ Ä !Ñ . ñ 8
Note: the sequence Ð+8 Ñ is actually a function 0 À Ä \ with the property that +œ0Ð8Ñ−FÐBÑ∩E8 " . Informally, the existence of such a function is completely clear. 8 But to be precise, this argument actually depends on the Axiom of Choice. The proof as written doesn't describe how to pick specific +8 's: it depends on making “arbitrary choices.” But using AC gives us a function 0 which “chooses” one point from each set in the collection ÖF" ÐBÑ ∩ E À 8 − ×Þ 8
Theorem 5.18 tells us something very important about the role of sequences in pseudometric spaces. The set EEœEE©E is closed iff cl . But cl is always true, so we can say E is closed iff clE©E . But that is true iff the limits of convergent sequences Ð+Ñ8 from E must also be in E . Therefore a complete knowledge about what sequences converge to what points in Ð\ß .Ñ would let you determine which sets are closed (and therefore, by taking complements, which sets are open). In other words, all the information about “which sets in Ð\ß .Ñ are open or closed?” is revealed by the convergent sequences. We summarize this by saying that in a pseudometric space Ð\ß .Ñß sequences are sufficient to describe the topology.
92 Example 5.19 If . is a pseudometric on \ , then .ww defined by . ( Bß CÑ œ min Ö"ß .ÐBß CÑ× is also a w pseudometric on \ . It is clear that . ÐB88 ß BÑ Ä ! iff .ÐB ß BÑ Ä ! . In other words, the metrics . and .Ð\ß.ÑÞw produce exactly the same convergent sequences and limits in Since sequences are sufficient to determine the topology in pseudometric spaces, we conclude that .. and w are equivalent pseudometrics on \Þ This example also shows that for any given Ð\ß .Ñ there is always an equivalent pseudometric .\w on for which all distances are Ÿ" : every pseudometric is equivalent to a bounded pseudometric. ww .ÐBßCÑ Another modification of ..ÐBßCÑœ that accomplishes the same thing is ".ÐBßCÑ . This time, it is a little harder to verify that ..ww is in fact a pseudometric the triangle inequality for ww ww wwww w takes a bit of work. Clearly . ÐBß CÑ Ÿ " , and . ÐB88 ß BÑ Ä ! iff .ÐB ß BÑ Ä ! . So . µ . µ . Þ
Definition 5.20 The diameter of a set E in Ð\ß .Ñ is defined by diam ÐEÑ œ sup Ö.ÐBß CÑ À Bß C − E× (we allow the possibility that diamÐEÑ œ ∞Ñ . If E has finite diameter, we say that E is a bounded setÞ
The diameter of a set depends on the particular metric used. Since we can always replace . by an equivalent metric ..www or for which diam Ð\ÑŸ" , boundedness is not a topological property .
It is an easy exercise to show that EE©FÐBÑ5Ð is bounded iff 5 ! for some sufficiently large where B\ÑÞ! can be any point in
The fact that the convergent sequences determine the topology in Ð\ß .Ñ gives us an upper bound on the size of certain metric spaces.
Theorem 5.21 If H is a dense set in a metric space Ð\ß .Ñ , then l\l Ÿ lHli! Þ In particular, for a i! separable metric space Ð\ß.Ñ , it must be true that l\lŸi! œ-Þ
Proof For each B−\ , pick a sequence Ð.Ñ88 in H such that Ð.ÑÄBÞ This sequence is actually a function0−HÞB Since a sequence in a metric space has at most one limit, the mapping i! FFÀ\ÄH given by ÐBÑœ0B is one-to-one, so l\lŸlHlœlHl Þ ñ
i! Note: If lHlœ7i! , do not jump to the (false) conclusion that l\lŸ7 œ7 . See the note of caution in Chapter I at the end of Example 14.8.
Theorem 5.21 is not true if Ð\ß .Ñ is merely a pseudometric space. For example, let \ be an uncountable set (with arbitrarily large cardinality) and let .\ be the trivial pseudometric on . Then any singleton ÖB× is dense, but l\l"i! œ" . In this case, where does the proof of Theorem 5.21 fall apart?
Sequences are sufficient to determine the topology in a pseudometric space, and continuity is characterized in terms of open sets, so it should not be a surprise that sequences can be used to decide whether or not a function 0 between pseudometric spaces is continuous.
93 Theorem 5.22 Suppose Ð\ß .Ñ and Ð] ß =Ñ are pseudometric spaces, and that 0 À \ Ä ] . Then 0 is continuous at + − \ iff Ð0ÐB88 ÑÑ Ä 0Ð+Ñ for every sequence ÐB Ñ Ä + .
Proof Suppose 0+ is continuous at and consider any sequence ÐBÑÄ+[8 . If is a neighborhood of " " 0Ð+Ñßthen 0 Ò[ Ó is a neighborhood of + , so ÐB88 Ñ is eventually in 0 Ò[ Ó . This implies that Ð0ÐB ÑÑ is eventually in [ , so Ð0ÐB8 ÑÑÄ 0Ð+ÑÞ
Conversely, if 0 is not continuous at + , then µ Ða%$ ! b ! 0ÒF$ ÐBÑÓ © F% Ð0Ð+ÑÑÑ , that is, b!a!0ÒFÐBÑÓ©%$ $ Î F% Ð0Ð+ÑÑ . " For this %$ and œß we have that 0ÒFÐBÑÓ©" Î F% Ð0Ð+ÑÑ . So for each 8 we can choose a point 8 8 " B8888 − F" ÐBÑ for which 0ÐB Ñ Â F% Ð0ÐBÑÑÞ Then ÐB Ñ Ä B (because .ÐB ß BÑ Ä ! ), 8 8 but Ð0ÐB88 ÑÑ ÄÎ 0Ð+Ñ (because =Ð0ÐB Ñß 0Ð+ÑÑ % for all 8 ). Therefore 0 is not continuous at + . ñ
Note: the first half of the proof is phrased completely in terms of neighborhoods of B! and 0ÐB! Ñà that part of the proof is topological. However the second half makes explicit use of the metric. In fact, as we shall see later, the second half of the proof must involve a little more than just the open sets.
Notice also that the second half of the proof makes uses the Axiom of Choice (the function BB8 “chooses” 8 for each .
The following theorem and its corollaries are often technically useful. Moreover, they show us that a pseudometric space Ð\ß .Ñhas lots of “built-in” continuous functions these functions can be defined using pseudometric . .
Theorem 5.23 In a pseudometric space Ð\ß .Ñ :
if ÐB88 Ñ Ä B and ÐC Ñ Ä Cßthen .ÐB 88 ß C Ñ Ä .ÐBß CÑ .
%% Proof Given % ! , pick R large enough so that 8 R implies both .ÐBßBÑ88## and .ÐCßCÑ Þ Since .ÐB88 ß C Ñ Ÿ .ÐB 8 ß BÑ .ÐBß C 8 Ñ Ÿ .ÐB 8 ß BÑ .ÐBß CÑ .ÐCß C 8 Ñ , we get that
%% if 8 R , .ÐB88 ß C Ñ ## .ÐBß CÑ œ .ÐBß CÑ % ЇÑ
Similarly, .ÐBß CÑ Ÿ .ÐBß B8888 Ñ .ÐB ß C Ñ .ÐC ß CÑ , so that
%% if 8 R , .ÐBß CÑ ## .ÐB88 ß C Ñ œ .ÐB 88 ß C Ñ % Ї‡Ñ
Combining Ð‡Ñ and Ї‡Ñ gives that l.ÐB88 ß C Ñ.ÐBß CÑl % if 8 R , so .ÐB 88 ß C Ñ Ä .ÐBß CÑ . ñ
Corollary 5.24 If +Ð\ß.ÑÐBÑÄB is a point in the pseudometric space and if 8 , then .ÐB8 ß +Ñ Ä .ÐBß +Ñ (in ‘ ).
94 Proof In Theorem 5.23, let ÐC88 Ñ be the constant sequence where C œ + for each 8 . ñ
Corollary 5.25 Suppose + − Ð\ß .Ñ . Define 0 À \ Ä‘ by 0ÐBÑ œ .ÐBß +ÑÞ Then 0 is continuous.
Proof Let B!8!8! be a point in \Þ If ÐB Ñ Ä B , then by Corollary 5.24, Ð.ÐB ß +ÑÑ Ä .ÐB ß +Ñ , that is Ð0ÐB8! ÑÑ Ä 0ÐB Ñ. So 0 is continuous at B ! (by Theorem 5.22 ). ñ
Recall that for a nonempty subset E of \ß we defined .ÐBß EÑ œ inf Ö.ÐBß +Ñ À + − E× . The following theorem is also useful.
Theorem 5.26 If EÐ\ß.Ñ is a nonempty subset of the pseudometric space , then the function 0À\Ä‘ defined by 0ÐBÑœ.ÐBßEÑ is continuous.
Proof We show that 0B−\+−E is continuous at each point ! . Let . Then the following inequalities are true for any B−\ :
.Ð+ß B!! Ñ Ÿ .Ð+ß BÑ .ÐBß B Ñ .Ð+ß BÑ Ÿ .Ð+ß B!! Ñ .ÐB ß BÑ
Apply “inf+−E ” to each inequality to get
.ÐEß B!!!! Ñ Ÿ .ÐEß BÑ .ÐBß B Ñ or .ÐEß B Ñ .ÐEß BÑ Ÿ .ÐBß B Ñ .ÐEß BÑ Ÿ .ÐEß B!! Ñ .ÐB ß BÑ or .ÐEß BÑ .ÐEß B ! Ñ Ÿ .ÐBß B ! Ñ so for all B−\ ,
l.ÐEß BÑ .ÐEß B!! Ñ | Ÿ .ÐBß B Ñ . In other words,
for all B − \, l0ÐBÑ 0ÐB!! Ñ | Ÿ .ÐBß B ÑÞ Ð* Ñ
So for %$%$ ! , we can choose œ . Then if .ÐBß B!! Ñ , we have l0ÐBÑ 0ÐB Ñl % Þ Therefore 0 is continuous at Bñ! .
Comments on the proof:
i) Given %$%! , the same choice œ can be used for every point B! . A function 0 that satisfies this condition stronger than mere continuity is called uniformly continuous . We will discuss uniform continuity more in Chapter IV. )
ii) From the last inequality (*) we could have argued instead: for any sequence ÐB8! Ñ Ä B ß we have.ÐB8! ß B Ñ Ä !, and this forces Ð0ÐB 8 ÑÑ Ä 0ÐB ! ÑÞ Therefore 0BÞ is continuous at ! However this argument hides the observation about “uniform continuity” made in i).
95 Exercises
E22. Suppose +−E©‘‘ and that 0ÀEÄ Þ We said that:
0+a!b!B−ElB+l is continuous at if %$ such that if and $ , then l0ÐBÑ 0Ð+Ñ % Þ
The order of the quantifiers is important. What functions are described by each of the following modifications of the definition:
a) a!b! $% such that if B−E and lB+l $ , then l0ÐBÑ0Ð+Ñ % b) a!a! %$ such that if B−E and lB+l $ , then l0ÐBÑ0Ð+Ñ % c) b!b! %$ such that if B−E and lB+l $ , then l0ÐBÑ0Ð+ÑÞ %
In each case, what happens if the restriction “! ” is dropped on either %$ or ?
E23. Suppose 0À‘‘ Ä Þ What does each of the following statements tell us about 0 ? (In this exercise, “interval” means “bounded open interval Ð+ß ,Ñ ”; and “ 0 takes M into N ” means that 0ÒMÓ © NÞ)
a) For every interval M+ containing and every interval N,0MN containing , takes into . b) There exists an interval N, containing and there exists an interval M+ containing such that 0MN takes into . c) There exists an interval N, containing such that for every interval M+0M containing , takes into N . d) There exists an interval N, containing such for every interval M+0 containing , does not take MN into . e) For every interval M+ containing , there exists an interval N,0M containing such that takes into N . f) There exists an interval M+ containing such that for every interval N,0M containing , takes into NÞ
E24. (The Pasting Lemma, easy version ) The two parts of the problem give conditions when a collection of continuous functions defined on subsets of \œ can be “united” ( “pasted together”) to form a new continuous function. In Ð\ß .Ñß suppose the sets Sα Ðα − EÑ are open and that J"8 ß ÞÞÞß J Ð8 − Ñ are closed.
a) Suppose that functions 0ÀSÄÐ]ß=Ñαα are continuous and that, if α" Á , then 0αα lÐS ∩ S " Ñ œ 0 " lÐS α ∩ S " Ñ(that is, 0 α and 0 " agree where their domains overlap). Then is continuous. αα−E0œ0Àαα −E SÄ]
b) Suppose that for each 3 œ "ß ÞÞÞß 8 ß 033 À J Ä Ð] ß =Ñ is continuous and that, if 3 Á 4 , then 0lJ33( ∩J 4 ) œ0lÐJ∩JÑ 43 4 (that is, 0 3 and 0 4 agree where their domains overlap). Then 88 is a continuous function. 0œ3œ" 033 À 3œ" J Ä]
c) Give an example to show that b) may be false for an infinite collection of functions 03 Ð3 − Ñdefined on closed subsets of \ , even if the domains J3 are pairwise disjoint.
96
E25. A point BÐ\ß.Ñ!8! in is a cluster point of the sequence ÐBÑ if for every neighborhood RB of and for all 8− ßb58 such that B5 −R . (When this condition is true, we say that “ÐB8 Ñ is frequently in every neighborhood of BÞ! ” ) Prove that if 0 À Ð\ß .Ñ Ä Ð] ß =Ñ is continuous and B!8 is a cluster point of ÐB Ñ in \ , then 0ÐB! Ñ is a cluster point of the sequence Ð0ÐB8 ÑÑ in ] .
"B−Eif E26. For E©‘; , its characteristic function is defined by E ÐBÑœ Þ !BÂEif For which sets EÀÄ is ;‘E ‘ continuous?
E27. Show that a set SÐ\ß.Ñ is open in if and only if there is a continuous function 0À\Ä‘ and an open set [ in ‘ such that S œ 0" Ò[ ÓÞ
E28. Suppose ÐB8 Ñ is a sequence in \ , that B is some point in \ and that Ð0ÐB8 ÑÑ Ä 0ÐBÑ for every 0−GÐ\Ñ. Prove that ÐBÑÄB8 or give a counterexample to show the statement is false.
w E29. Let ..ÐBÑÄ!. be the usual metric on ‘‘ . Find a metric on such that 8 with respect to iff ww ÐB8 Ñ Ä ! with respect to . , but . is not equivalent to . .
E30. a) Suppose EÐ\ß.ÑBÂE is a closed set in the pseudometric space and that ! . Prove that there is a continuous function 0 À \ Ä Ò!ß "Ó such that 0lE œ ! and 0ÐB! Ñ œ "Þ (Hint: Consider the function “distance to the set E .”)
b) Suppose EF and are disjoint closed sets in Ð\ß.Ñ . Prove that there exists a continuous .ÐBßEÑ function 0À\Ä‘ such that 0lEœ! and 0lFœ" . ÐHint: Consider .ÐBßEÑ .ÐBßFÑ Ñ
c) Using b) (or by some other method ) prove that if EF and are disjoint closed sets in Ð\ß.Ñ , then there exist open sets YZ and such that E©YßF©ZY∩ZœgÞYZ and Can and always be chosen so that clY∩ cl Z œg ?
E31. E0ÀÐ\ß.ÑÄ]= function ( , ) is called an isometry between \ and ] if 0 is onto and, for all Bß C − \, .ÐBß CÑ œ =Ð0ÐBÑß 0ÐCÑÑ . If such an 0 exists, we say that Ð\ß .Ñ and Ð] ß =Ñ are isometric to each other. If 00\]0 is not onto, we say is an isometry of into , or that is an isometric embedding of Ð\ß .Ñ into Ð] ß =Ñ. Let ‘‘ and # have their usual metrics.
a) Prove that there is no isometry between ‘‘ and # . b) Let +−‘‘‘ . Prove that there are exactly two isometries from onto which hold the point + fixed (that is, for which 0Ð+Ñ œ + ). c) Give an example of a metric space which is isometric to a proper subset of itself.
97 E32. Use convergent sequences to prove the theorem that two continuous functions 01 and from Ð\ß .Ñ into a metric space are identical if they agree on a dense set in \ .
E33. Suppose 0 À‘‘ Ä . Then we can define J À ‘‘ Ä# by J ÐBÑ œ ÐBß 0ÐBÑÑ , so that ran ÐJ Ñ is the graph of 0 .
a) Prove that the following statements are equivalent: i) 0 is continuous ii) J is continuous iii) The sets ÖÐBß CÑ À C 0ÐBÑ× and ÖÐBß CÑ À C Ÿ 0ÐBÑ× are both closed in ‘# .
b) Prove that if 0 is continuous, then its graph is a closed set in ‘# . Give a proof or a counterexample for the converse.
E34. Suppose \ œÖBßCßDßA× is a four point set.
a) Show that the information sszsÐBß CÑ œ ÐCß Ñ œ ÐDß BÑ œ # and ssÐBß AÑ œ sÐCß AÑ œ ÐDß AÑ œ " determines a unique metric s on \ß i.e., that there is one and only one metric =\ on which satisfies the given conditions. b) Show that Ð\ßs Ñ cannot be isometrically embedded into the plane ‘# (with its usual metric). c) Prove or disprove: Ð\ß =Ñ can be isometrically embedded in Ðj# ß .Ñ , where . is the usual metric on j# .
E35. Suppose Ð\ß .Ñ is a metric space for which l\l " and in which g and X are the only clopen sets. Prove that l\l - . (Hint: First prove that there must be a nonconstant continuous function 0À\Ä‘ . What can you say about the range of 0 ? )
E36. Let \GÐ\Ñ be a finite set and let ‡ be the set of all bounded continuous functions from \ into ‘. Let sC denote the “uniform metric” on ‡Ð\Ñ given by =Ð0ß 1Ñ œ sup Öl0ÐBÑ 1ÐBÑl À B − \×Þ
a) Prove that ÐG‡ Ð\Ñß =Ñ is separable. b) If \œ , is part a) still true?
98 Chapter II Review
Explain why each statement is true, or provide a counterexample.
1. A finite set in a metric space must be closed.
2. For 7ß 8 − ß write 7 8 œ #5 Dß where D is an integer not divisible by 2. Define .Ð7ß 7Ñ œ ! and ß for 7 Á 8ß .Ð7ß 8Ñ œ 5 . Then . is a pseudometric on .
3lB Cl 3. Consider the set Ò"ß∞Ñ with the metric =ÐBßCÑœ33lBCl . Let have its usual metric . and define0ÀÒ"ß∞ÑÄ by 0ÐBÑœ “the largest integer ŸB ”. Then 0 is continuous.
4. If RBÐ\ß.Ñ∩BÞ"# and NNN are neighborhoods of in , then "# is also a neighborhood of
5. For any open subset S of a metric space Ð\ß .Ñ , int Ð cl ÐSÑÑ œ SÞ
11 6. The metric .Ð8ß 7Ñ œ | 87 | on is equivalent to the usual metric on .
7. Define 0À‘ Ä by 0Ð8Ñœ the 8>2 digit of the decimal expansion of 1 . Then 0 is continuous.
8. The set of all real numbers with a decimal expansion of the form B œ !ÞB""$ B B ÞÞÞB 8 !"!"!"ÞÞÞ is dense in Ò!ß "ÓÞ
9. There are exactly - countable dense subsets of ‘ .
lBC l 10. Suppose we measure distances in ‘ using the metric .ÐBß CÑ œ "lBCl . Then the function cosÀÄ‘‘ is continuous at every point +− ‘ .
11. In a pseudometric space Ð\ß .Ñ , a subset E is dense if and only if int Ð\ EÑ œ g .
12. Let denote the “taxicab” metric 8 | on 88 . is dense in ( 8 . .>>33> .ÐBßCÑœ3œ" lB C‘ ‘ ß.Ñ 13. If FF is a countable subset of ‘‘ , then is dense in ‘ .
14. If E © Ò!ß "Ó and cl E Á Ò!ß "Ó , then int E Á g .
1 15. Let EœÖ8 : 8− }. The discrete unit metric produces the same topology on E as the usual metric.
16. In a pseudometric space Ð\ß .Ñ , cl E œ cl Ð\ EÑ if and only if A is clopen.
17. If UU is an open set in ‘ and ªYœ , then ‘ .
18. There is a sequence of open sets in such that ∞ SSœÞ88‘ 8œ"
19. If E©Ð\ß.Ñ , then int Eœ\ cl ( \EÑ .
99 20. There are exactly c continuous functions 0À Ä Þ
||B C n# 21. Let ‘‘ have the metric .ÐBß CÑ œ1|BC | and let +88 œn# +1 . In Ð ß .Ñ , (a ) Ä 1 .
22. Suppose .. is the usual metric on ‘ and " is another metric with the property that for every sequence Ð<8 Ñ À Ð<8"8 Ñ Ä & with respect to . if and only if Ð< Ñ Ä & with respect to .Þ Then .µ.Þ"
23. Suppose E©F©Ð\ß.Ñ . If cl Eœ cl F , int Eœ int F , and Fr Eœ Fr F , then EœF .
24. Suppose GÐÒ!ß "ÓÑ has the metric .Ð0ß 1Ñ œ" |fg | and define F‘ À ÐGÐÒ!ß "ÓÑß .Ñ Ä by ! FFÐ0Ñ œ" 0 . Then is continuous. ! 25. Let .Ðß.Ñß< be the trivial pseudometric on ‘‘ . In each real number is the limit of a sequence of irrational numbers.
26. Suppose 0À‘‘ Ä and 1À ‘‘ Ä are continuous and that 0Á1 . Then there must exist a point : − where 0Ð:Ñ Á 1Ð:Ñ and a point ; − where 0Ð;Ñ Á 1Ð;ÑÞ
27. In a pseudometric space Ð\ß .Ñ , cl E œ int E if and only if E is clopen.
28. If Ð\ß .Ñ is a metric space in which every convergent sequence is eventually constant, then gc. œÐ\Þ)
29. Let B − Ð\ß .Ñ Suppose ÐB88 Ñ is a sequence such that Ð0ÐB ÑÑ Ä 0ÐBÑ for every continuous 0À\Ä‘. Then ÐBÑÄB8 .
30. Let .‡ Ð0ß 1Ñ œ sup Ö | 0ÐBÑ 1ÐBÑl À B − Ò!ß "Ó× for 0ß 1 − GÐÒ!ß "ÓÑ . Let 0 − GÐÒ!ß "ÓÑ be the function given by 0ÐBÑ œ B # and let Ð08 Ñ be a sequence such that ‡ Ð08 Ñ Ä 0 in ÐGÐÒ!ß "ÓÑß . Ñ Then there is an R− such that 08 ÐBÑ B for all B−Ò!ß"Ó and all 8 R .
31. If 0À‘‘ Ä is continuous, then the graph of 0 is a closed subset of ‘# .
32. If the graph of a function 0À‘‘ Ä is closed subset of ‘# , then 0 is continuous.
33. The are exactly -. different metrics on ‘g for which . œ the usual topology on ‘ .
34. In ‘ , the interval Ò #ß "Ó can be written as a countable intersection of open sets.
35. For any 0−‘‘ ß then there is a continuous function 1− such that 1l œ0Þ
36. Let .. be the usual metric on ‘‘ and ww the discrete unit metric. Suppose 0ÀÐß.ÑÄÐß. ) is continuous. Then 0 is constant.
37. If l\l "ß then there are infinitely many different metrics . on \ for which g. is the discrete topology.
100 38. Suppose Ð\ß .Ñ is a pseudometric space and E © \ . E is dense in \ if and only if intÐ\ EÑ œ g .
39. The space of irrational numbers is separable.
40. Suppose Ð\ß .Ñ and Ð] ß .w Ñ are pseudometric spaces and that 0 À \ Ä ] is continuous at +Þ If .Ð+ß ,Ñ œ !, then 0 is also continuous at , .
41. If E and F are subsets of Ð\ß .Ñ , then int ÐE ∩ FÑ œ int ÐEÑ ∩ int ÐFÑ .
‡ 42. Let .. be the “max metric” on ‘‘‘ and let > be the “taxicab metric” on . For B− , let $‡ 0ÐBÑ œcos ( B ). Then 0 À Б‘ ß. Ñ Ä Ð ß.> Ñ is continuous.
43. Let . be a pseudometric on the set \ œ Ö!ß "× . Then either ggc.. œ Ögß \× or œ Ð\Ñ .
44. Suppose 0 À‘‘ Ä is continuous and 0Ð:Ñ œ : # for each irrational : . Then 0Ð"(Ñ − Þ 45. A finite set in a pseudometric space must be closed.
46. If 0À‘ Ä is continuous and 0Ð"Ñœ" , then there must exist an irrational number B for which 0ÐBÑ œ ".
47. In a metric space Ð\ß .Ñ , it cannot happen that F%% Ð+Ñ œ \ F Ð,Ñ .
48. The discrete unit metric produces the same topology on as the usual metric on .
49. If DCDC is an uncountable dense subset of ‘‘ and is countable, then is dense in .
50. Suppose that 0Àj# Ä‘ is continuous and that 0ÐBÑœ!for whenever B is any sequence in j# that "" " is eventually 0. Then 0ÐÐ"ß#$ ß ß ÞÞÞß 8 ß ÞÞÞ ÑÑ œ !Þ
51. If 0 À Ð\ß .Ñ Ä Ð] ß .w Ñ is continuous and onto, and \ is separable, then ] is separable.
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