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 ßthen S " ∩ ÞÞÞ ∩ S 8 − Þ A set S©\ is called open if S−gg. The pair Ð\ß Ñis 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 Ñis 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+−EßJααα−E iii) if are closed, then 8 is closed. J"8 ß ÞÞÞß J3œ3 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. œthat 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. œÞSo a trivial topological space turns out to be pseudometrizable. At the opposite extreme, suppose gcœÐ\Ñ. Then gis 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!× −.. ßso 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 Ñand 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 −EÞαα−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 g3œ" 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..ßÁÞso 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 Ñmetrizable 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 ßthat 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.