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Chapter 4: Completeness and Compactness

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Chapter 4: Completeness and Compactness Chapter IV Completeness and Compactness 1. Introduction In Chapter III we introduced topological spaces as a generalization of pseudometric spaces. This allowed us to see how certain ideas continuity, for example can be extended into a setting where there is no “distance” between points. Continuity does not really depend on the pseudometric . but only on the topology. Looking at topological spaces also highlighted the particular properties of pseudometric spaces that were really important for certain purposes. For example, it turned out that first countability is the crucial ingredient for proving that sequences are sufficient to describe a topology, and that Hausdorff property, not the metric ., is what matters to prove that limits of sequences are unique. We also looked at some properties that are distinct in topological spaces but equivalent in the special case of pseudometric spaces for example, second countability and separability. Most of the earlier definitions and theorems are just basic “tools” for our work. Now we look at some deeper properties of pseudometric spaces and some significant theorems related to them. 2. Complete Pseudometric Spaces Definition 2.1 A sequence ÐBÑ8 in a pseudometric space Ð\ß.Ñ is called a Cauchy sequence if a !b% N − such that if 7ß8 R, then .ÐBßBÑ7 8 % . Informally, a sequence ÐB8 Ñ is Cauchy if its terms “get closer and closer to each other.” It should be intuitively clear that this happens if the sequence converges, as the next theorem confirms. Theorem 2.2 If ÐBÑÄB8 in Ð\ß.Ñ, then ÐBÑ 8 is Cauchy. % Proof Let % !. Because ÐBÑÄB8 , there is an R− such that .ÐBßBÑ8 # for 8 R . % % So if 7ß8 R, we get .ÐBßBÑŸ.ÐBßBÑ.ÐBßBÑ7 8 7 8 # # œ%. Therefore ÐBÑ8 is Cauchy. ñ However, a Cauchy sequence does not always converge. For example, look at the space Ð ß.Ñ where . is the usual metric. Consider a sequence in that converges to È#in ‘ à for example, we could use Ð;8 Ñ œ Ð"ß "Þ%ß "Þ%"ß "Þ%"%ßÞÞÞ Ñ . ñ Since Ð;Ñ8 is convergent in ‘ ßÐ;Ñ 8 is a Cauchy sequence in ( “Cauchy” depends only on . and the numbers ;8, not whether we are thinking of these ; 8 'sas elements of or ‘ . ñ But Ð;Ñ8 has no limit in Ðß.Ñ . (Why? To say that “È#  ” is part of the answer. ) 154 Definition 2.3 A pseudometric space Ð\ß .Ñ is called complete if every Cauchy sequence in Ð\ß.Ñhas a limit in \ . Example 2.4 In all parts of this example, . is the usual metric on subsets of ‘. 1) Б ß.Ñ is complete ( for the moment, we assume this as a simple fact from analysis; however we will prove it soon ) but Ð ß.Ñ and Ðß.Ñ are not complete: completeness is not a hereditary property ! 2) If ÐBÑ8 is a Cauchy sequence in Ðß.Ñ , then there is some R such that lBBl7 8 1 for 7ß8 R. Because the B8's are integers, this implies that BœB7 8 for 7ß8 R that is, ÐB8 Ñ is eventually constant and therefore converges. Therefore Ð ß.Ñ is complete. w " " w 3) Consider with the equivalent metric .Ð7ß8Ñœl7 8 l . ( . µ . because both metrics produce the discrete topology on .) w w In Ð ß.Ñ , the sequence ÐBÑœÐ8Ñ8 is Cauchy, and if 5−, then .ÐBß5Ñ8 " " " w œllÄ8 5 5 Á!as 8Ä∞ , so ÐBÑÄ58 / . So the space Ðß.Ñ is not complete. Since Ðß.Ñ w has the discrete topology, the map 3Ð8Ñœ8 is a homeomorphism between Ðß.Ñw and the complete space ( ß.ÑÞ So completeness is not a topological property . A homeomorphism takes convergent sequences to convergent sequences and nonconvergent sequences to nonconvergent sequences. For example, Ð8Ñ does not converge in Ðß.Ñ w and Ð3Ð8ÑÑœÐ8Ñ does not converge in Ðß.Ñ . These two homeomorphic metric spaces have exactly the same convergent sequences but they do not have the same Cauchy sequences . Changing a metric . to a equivalent metric . w does not change the open sets and therefore does not change which sequences converge. But the change might create or destroy Cauchy sequences because the Cauchy property does depend specifically on how distances are measured. Another similar example: we know that any open interval Ð+ß ,Ñ is homeomorphic to ‘ . However, with the usual metric on each space, ‘ is complete and Ð+ß ,Ñ is not complete. 4) In light of the observations in 3) we can ask: if Ð\ß .Ñ is not complete, might there be some equivalent metric .w for which Ð\ß.Ñ w is complete? To take a specific example: could we find some metric .w on so .µ. wbut for which Ðß.Ñ w is complete? We could also ask this question about Ð ß .ÑÞ Later in this chapter we will see the answer and perhaps surprisingly, the answers for and are different x w " 5) Ðß.Ñ and ÐÖ8 À8− ×ß.Ñ are countable discrete spaces, so they are " homeomorphic. But (even better!) the function 0Ð8Ñ œ 8 is actually an isometry between these w " " spaces because . Ð7ß8Ñœl7 8 lœ.Ð0Ð7Ñß0Ð8ÑÑÞ An isometry is a homeomorphism, but since it preserves distances, an isometry also takes Cauchy sequences to Cauchy sequences and non-Cauchy sequences to non-Cauchy sequences. Therefore if there is an isometry between two pseudometric spaces, then one space is complete iff the other space is complete. w For example, the sequence ÐBÑœ88 is Cauchy in Ðß.Ñ and the isometry 0 carries " " ÐBÑ8 to the Cauchy sequence Ð0ÐBÑÑœÐÑ8 8 in ÐÖÀ8−×ß.Ñ 8 . It is clear that Ð8Ñ has no limit " in the domain and Ð8 Ñ has no limit in the range. w " Ðß.Ñ and ÐÖÀ8−×ß.Ñ8 “look exactly alike” not just topologically but as metric spaces. We can think of 0 as simply renaming the points in a distance-preserving way. 155 Theorem 2.5 If B is a cluster point of a Cauchy sequence ÐBÑÐ\ß.Ñ8 in , then ÐBÑÄB 8 . % Proof Let % !. Pick R so that .ÐBßBÑ7 8 # when 7ß8 RÞ Since ÐBÑ8 clusters at B , we can pick a O R so that B−FÐBÑ% . Then for this O and for 8 Rß O # % % .ÐB8 ßBÑŸ.ÐB 8O ßB Ñ.ÐB O ßBÑ# # œ % so ÐB8 ÑÄB. ñ Corollary 2.6 In a pseudometric space Ð\ß.Ñ: a Cauchy sequence ÐBÑ8converges iff ÐBÑ 8 has a cluster point. Corollary 2.7 In a metric space Ð\ß .Ñ , a Cauchy sequence can have at most one cluster point. Theorem 2.8 A Cauchy sequence ÐBÑ8 in Ð\ß.Ñ is bounded that is, the set ÖB" ßB # ßÞÞÞßB 8 ßÞÞÞ× has finite diameter. Proof Pick R so that .ÐB7, BÑ" 8 when 7ß8 R. Then B−FÐBÑ8 " R for all 8 RÞ Let <œmax Ö"ß.ÐBßBÑß.ÐBßBÑßÞÞÞß.ÐB"R #R R"R ßBÑ×Þ Therefore, for all 7ß8 we have .ÐB78 ßB ÑŸ.ÐB 8R ßB Ñ.ÐB R7 ßB ÑŸ#<, so diam ÖBßBßÞÞÞßB "#8 ßÞÞÞן#<Þ ñ Now we will prove that ‘ (with its usual metric .) is complete. The proof depends on the completeness property (also known as the “least upper bound property”) of ‘. We start with two lemmas which might be familiar from analysis. A sequence ÐB8 Ñ is called weakly increasing if BŸB8 8" for all 8, and weakly decreasing if B B8 8" for all 8 . A monotone sequence is one that is either weakly increasing or weakly decreasing. Lemma 2.9 If ÐB8 Ñ is a bounded monotone sequence in ‘, then ÐB8 Ñ converges. Proof Suppose ÐBÑ8 is weakly increasing. Since ÐBÑ8 is bounded, ÖBÀ8−× 8 has a least upper bound in ‘. Let Bœsup ÖBÀ8−8 × . We claim ÐBÑÄBÞ 8 Let %% !. Since BBß we know that B % is not an upper bound for ÖBÀ8−×8 . Therefore BBŸB% R for some RÞSince ÐBÑ8 is weakly increasing and B is an upper bound for ÖBÀ8−×ß8 it follows that BBŸBŸB % R 8 for all 8 R . Therefore lBBlŸ8 % for all 8 Rß so ÐBÑÄB8 . If ÐB8 Ñ is weakly decreasing, then the sequence ÐB8 Ñ is a weakly increasing, so ÐBÑÄ+8 for some +; then ÐBÑÄ+Þñ 8 Lemma 2.10 Every sequence ÐB8 Ñ in ‘ has a monotone subsequence. Proof Call B5a peak point of ÐBÑB B 858if for all 8 5 . We consider two cases. i) If ÐB8 Ñ has only finitely many peak points, then there is a last peak point B8! in ÐBÑ 8 . Then B8 is not a peak point for 8 8Þ! Pick an 8 8 "!. Since B 8 " is not a peak point, there is an 8 8# " with BB 8"## 8. Since B 8 is not a peak point, there is an 8 8$# with BB 8# 8 $ . We continue in this way to pick an increasing subsequence ÐB85 Ñof ÐBÑ 8 . 156 ii) If ÐB8 Ñ has infinitely many peak points, then list the peak points as a subsequence B88ß" ßB # ÞÞÞ (where 88ÞÞÞÑ "# . Since each of these points is a peak point, we have B B85 8 5" for each 5ÐBÑ, so 8 5 is a weakly decreasing subsequence of ÐBÑñ8 . Theorem 2.11 Ð‘ß .Ñ is complete. Proof Let ÐBÑ8 be a Cauchy sequence in Ðß.Ñ‘ . By Theorem 2.8, ÐBÑ8 is bounded and by Lemma 2.10, ÐBÑ8 has a monotone subsequence ÐBÑ8 5 which is, of course, also bounded. By Lemma 2.9, ÐBÑ85 converges to some point B−‘. Therefore B is a cluster point of the Cauchy sequence ÐBÑ8, so ÐBÑÄBÞ 8 ñ Corollary 2.12 Б8ß .Ñ is complete.
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