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Chapter IV Completeness and Compactness

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Chapter IV Completeness and Compactness Chapter IV Completeness and Compactness 1. Introduction In Chapter III we introduced topological spaces a generalization of pseudometric spaces. This allowed us to see how certain ideas continuity, for example can be extended into a situation 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 equivalent in pseudometric spaces for example, second countability and separability but are not equivalent in topological spaces. 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 ÐB8 Ñ in a pseudometric space Ð\ß .Ñ is called a Cauchy sequence if a!b% MNß − such that if 7ß8R, then .ÐBßBÑ78 %. 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, and the next theorem confirms this. Theorem 2.2 If ÐB88 Ñ Ä B in Ð\ß .Ñ, then ÐB Ñ is Cauchy. % Proof Let % !. Because ÐB88 Ñ Ä B, there is an R − such that .ÐB ß BÑ # for 8 R. %% So if 7ß 8 R, we get .ÐB78 ß B Ñ Ÿ .ÐB 7 ß BÑ .ÐBß B 8 Ñ ## œ%. Therefore ÐB 8 Ñ is Cauchy. ñ A Cauchy sequence does not always converge. For example, look at the space Ðß.Ñ where . is the usual metric. Consider a sequence in ‘ that converges to #àin for example, we could used Ð;888 Ñ œ Ð"ß "Þ%ß "Þ%"ß "Þ%"%ß ÞÞÞ Ñ. Since Ð; Ñ is convergent in ‘ ß Ð; Ñ is a Cauchy sequence in (“Cauchy” depends only on .; and the numbers 8, not whether we are thinking of these ;8's as elements of ‘ or ..) But Ð;8 Ñ has no limit in . (Why? To say that “# ” 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 .Ñ has a limit in \. Example 2.4 Throughout this example, . denotes the usual metric on subsets of ‘. 1) Ðß.Ñ‘ is complete. (For the moment, we take this as a simple fact from analysis; however we will prove it soon.) But Ð ß .Ñ and Ð ß .Ñ are not complete À completeness is not hereditary! 2) Let ÐB8 Ñ be a Cauchy sequence in Ð ß .Ñ. Then there is some R such that lB78 B l 1 for 7ß 8 R. But the B 8's are integers, so this means that B 7 œ B 8 for 7ß 8 R. So every Cauchy sequence is eventually constant and therefore converges. Ð ß .Ñ is complete. "" 3) Consider with the metric ."8 Ð7ß 8Ñ œ l78 l. Then the sequence ÐB Ñ œ Ð8Ñ is "" " Cauchy. If 5−, then .ÐBß5Ñœl"885 lÄ 5 Á!as 8Ä∞ , so ÐBÑ 8does not converge to 5Ðß.Ñ. The space " is not complete. However Ðß.Ñ " has the discrete topology, so the identity function 0Ð8Ñœ8 is a homeomorphism between Ðß.Ñ" and the complete space ( ß.ÑÞ So completeness is not a topological property. A homeomorphism takes convergent sequences to convergent sequences and non convergent sequences to nonconvergent sequences. For example, Ð8Ñ does not converge in Ð ß ." Ñ and Ð0Ð8ÑÑ does not converge in Ð ß .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 topologically equivalent metric w does not change the open sets and does not change which sequences converge. But the change may create or destroy Cauchy sequences since this property depends on distance measurements. Another similar example: we know that any open interval Ð+ß ,Ñ is homeomorphic to ‘ . But Ð+ß ,Ñ is not complete and ‘ is complete (where both space have the metric .). 4) In light of the comments in 3) we can ask: if Ð\ß .Ñ is not complete, might there be a different metric .µ.ww for which Ð\ß.Ñ is complete? To be specific: could we find a metric .w on so . µ.wwbut where Ð ß.Ñ is complete? We could also ask this question about Ð ß.ÑÞ Later in this chapter we will see that the answer is “no” in one case but “yes” in the other. " 5) Ðß.Ñ" and ÐÖÀ8−×ß.Ñ8 are both countable discrete spaces, so they are " homeomorphic. In fact, the function 0Ð8Ñ œ8 is an isometry between these spaces: ." Ð7ß8Ñ "" œ|78 l œ .Ð0Ð7Ñß 0Ð8ÑÑÞ 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. For example, the sequence ÐB8" Ñ œ 8 is Cauchy is Ð ß . Ñ and the isometry 0 carries "" ÐB88 Ñ to the Cauchy sequence Ð0ÐB ÑÑ œ Ð88 Ñ in ÐÖ À 8 − ×ß .Ñ. It is clear that Ð8Ñ has no limit " in the domain and ÐÑ8 has no limit in the range. " 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. Theorem 2.5 If BÐBÑÐ\ß.ÑÐBÑÄB is a cluster point of a Cauchy sequence 88 in , then . 155 % Proof Let % !. Pick R so that .ÐB78 ß B Ñ # when 7ß 8 RÞ Since ÐB 8 Ñ clusters at B, we can pick a OR so that B −FÐBÑ% . Then for this O and for 8Rß O # %% .ÐB88OO ß BÑ Ÿ .ÐB ß B Ñ .ÐB ß BÑ ## œ % so ÐB8 Ñ Ä B. ñ Corollary 2.6 A Cauchy sequence ÐB8 Ñ in Ð\ß .Ñ converges iff it 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 ÐB8"#8 Ñ in Ð\ß .Ñ is bounded that is, the set ÖB ß B ß ÞÞÞß B ß ÞÞÞ× has finite diameter. Proof Pick R so that .ÐB78, B Ñ " when 7ß 8 R. Then B 8 − F "R ÐB Ñ for all 8 RÞ Let < œmax Ö"ß .ÐB"R ß B Ñß .ÐB #R ß B Ñß ÞÞÞß .ÐB R"R ß B Ñ×Þ Therefore, for all 7ß 8 we have .ÐB78 ß B Ñ Ÿ .ÐB 8R ß B Ñ .ÐB R7 ß B Ñ Ÿ #<, so diam ÖB "# ß B ß ÞÞÞ ß B 8 ß ÞÞÞ× Ÿ #<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 BŸB88"for all 8, and weakly decreasing if B B 88" for all 8. A monotone sequence is one that is either weakly increasing or weakly decreasing. Lemma 2.9 If ÐB88 Ñ is a bounded monotone sequence in ‘, then ÐB Ñ converges. Proof Suppose ÐB888 Ñ is weakly increasing. Since ÐB Ñ is bounded, ÖB À 8 − × has a least upper bound in ‘. Let Bœsup ÖB88 À8− × . We claim ÐBÑÄBÞ Let %%!. Since B Bß we know that B % is not an upper bound for ÖB8 À8− ×. Therefore B% BR8 ŸB for some RÞSince ÐBÑ is weakly increasing and B is an upper bound for ÖB8R88 À8−% ×ß it follows that B B ŸB ŸB for all 8 R. Therefore lB BlŸ % for all8 Rß so ÐBÑÄB8 . If ÐB88 Ñ is weakly decreasing, then Ð B Ñ is a weakly increasing sequence so ÐBÑÄ+88 for some +; then ÐBÑÄ+Þñ Lemma 2.10 Every sequence ÐB8 Ñ in ‘ has a monotone subsequence. Proof Call BÐBÑB B8 55858a peak point of if for all . We consider two cases. i) If ÐB8 Ñ has only finitely many peak points, then there is a last peak point B88! in ÐB Ñ. Then B8!"!8 is not a peak point for 88Þ Pick an 8 8. Since B " is not a peak point, there is an 8#" 8 with B 8"# B 8. Since B 8 # is not a peak point, there is an 8 $# 8 with B 8#$ B 8. We continue in this way to pick an increasing subsequence ÐB885 Ñof ÐB Ñ. ii) If ÐB8 Ñ has infinitely many peak points, then list the peak points as a subsequence B88ß"# ß B ÞÞÞ (where 8 " 8 # ÞÞÞÑ. Since each of these points is a peak point, we have B8855" B for each 5ÐBÑ, so 885 is a weakly decreasing subsequence of ÐBÑñ. 156 Theorem 2.11 Бß.Ñ is complete. Proof Let ÐB8 Ñbe a Cauchy sequence in Б ß .Ñ. By Theorem 2.8, ÐB8 Ñ is bounded and by the Lemma 2.10, ÐBÑ888 has a monotone subsequence ÐBÑÞ55Of course ÐBÑ is also bounded.
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