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Some Examples and Counterexamples of Advanced Compactness in Topology

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Some Examples and Counterexamples of Advanced Compactness in Topology International Journal of Research in Engineering and Science (IJRES) ISSN (Online): 2320-9364, ISSN (Print): 2320-9356 www.ijres.org Volume 9 Issue 1 ǁ 2021 ǁ PP. 10-16 Some Examples and Counterexamples of Advanced Compactness in Topology Pankaj Goswami Department of Mathematics, University of Gour Banga, Malda, 732102, West Bengal, India ABSTRACT: The examples and counter examples are always usefull for better comprehension of underlying concept in a theorem or definition .Compactness has come to be one of the most importent and useful topic in advanced mathematics.This paper is an attempt to fill in some of the information that the standard textbook treatment of compactness leaves out, and giving some constructive significative counterexamples of Advanced Compactness in Topology . KEYWORDS: open cover, compact, connected, topological space, example, counterexamples, continuous function, locally compact, countably compact, limit point compact, sequentially compact, lindelöf , paracompact --------------------------------------------------------------------------------------------------------------------------------------- Date of Submission: 15-01-2021 Date of acceptance: 30-01-2021 --------------------------------------------------------------------------------------------------------------------------------------- I. INTRODUCTION We begin by presenting some definitions, notations,examples and some theorems that are essential for concept of compactness in advanced topology .For more detailed, see [1-2], [7-10] and others, in references .If X is a closed bounded subset of the real line ℝ, then any family of open sets in ℝ whose union contains X has a finite sub family whose union also contains X . If X is a metric space or topological space in its own right , then the above proposition can be thought as saying that any class of open sets in X whose union is equal to X has a finite subclass whose union is also equal to X. The topological space for which the conclusion of this theorem holds are known as compact topological space. In particular that collection of open sets is called as an open cover. Defination 1.1. Let X be a topological space and A be a subset of X. An open cover of A is collection = {Gi} of open sets in X whose union contains A; that is, A subcollection ʹ of whose union is also contains A; that is, is called as subcover of A in . Example 1.2. Let A = (0,1) ⊂ ℝ then for each n ℕ , Gn = ( is an open cover of A that is . Defination 1.3. : A topological space X is said to be compact if every open covering of X contains a finite subcover that also covers X. Theorem 1.4.A subser A of R is compact iff A is bounded and closed.This is known as Heine- Borel theorem. In particular a subset of ℝ is compact iff it is closed and bounded. The proof is given in [10] and also in others. Example 1.5. 1. Any closed bounded subset [a, b] ⊂ ℝ is compact by Heine –Borel theorem. 2.The real line ℝ is not compact,for it, each n ℕ ,Gn = (-n, n) is an open cover which has no finite sub cover. 3. A = (0,1) ⊂ ℝ then for each n ℕ , Gn = ( is an open cover of A that is , clearly it has no finite sub cover, so A is not compact. In particular for any open interval A = (a, b) ⊂ ℝ , for each n ℕ ; Gn = (a -ϵn, b − ϵn),ϵn = ,where c be any point in (a,b); is an open cover of (a, b) which has no finite subcover. www.ijres.org 10 | Page Some Examples and Counterexamples in Advanced Compactness of Topology 4.Any finite discrete topological space is compact.As there are only finite number of open sets. 5.Any set X with cofinite topology is compact. For it , if U is open in X ,then X-U is finite and so X-U can be covered by a finite number of open sets say G1, G2, ..., Gn . Then for the collection Cn= {U, G1, G2, ..., Gn } is a open cover for X which is also finite. Hence any set X with cofinite topology is compact. 6.Cantor set is compact scince it is closed bounded subset of ℝ . Theorem 1.6. Every closed subset of compact space is compact. Proof is given in [10] and also in others. Theorem 1.7. Continuous image of compact spaces is compact. Proof is given in [10] and also in others. Definition1.8. A topological space X is said to be sequentially compact iff every sequence in X has a convergent sub-sequence. Example 1.9. 1.The closed interval [a,b] in ℝ is sequentially compact ,since any sequence in [a,b] is bounded ,and thus has a convergent subsequence.In particular compact metric spaces is sequentially compact. 2.The real line ℝ is not sequentially compact.The sequence {n} has no convergent sub-sequence.So in particular ℝ is not sequentially compact. 3.Any finite discrete space compact so sequentially compact but infinite discrete space is not sequentially compact. 4.The interval (0,1) is not sequentially compact .The sequence { } has no convergence sub-sequence in (0,1). Definition1.10. A topological space X is said to be locally compact iff every point in X has at least one nbd whose closure is compact. Example 1.11. 1.The real line ℝ is locally compact .For any point x in R there is an open interval (a,b) containing x whose closure [a,b] is compact. 2.In particular ℝ is locally compact. 3.The metric space ℚ set of all rational numbers with usual metric is not locally compact.For it closed intervals [a,b]∩ ℚ in ℚ are not compact ,it follows that all compact subsets of ℚ have empty interior.Hence ℚ is in locally compact. 4.ℝ ,ℝ with lower limit topology constructed from the interval {[a,b): a<b} is not locally compact as every compact subset of ℝ is countable , so has empty interior . 5.Any discrete metric space is locally compact,science for each point has a ndb whose closure is compact. Definition1.12. A topological space X is said to be countably compact iff every countable open cover of X has a finite subcover. Example 1.13. 1. Any compact spaces are the example of countably compact space. Definition1.14. A topological space X is said to be limit point compact(Bolzano- Weierstrass Property ) BWP if each infinite subset of X has a limit point in X. Example 1.15. 1.Every closed bounded interval [a,b] of ℝ , is limit point compact as each infinite subset of it has a limit point in it. 2.ℝ is not limit point compact ,as set of integers has no limit point on it. 3.In particular any compact spaces are limit point compact. Definition1.16. A collection = {Gi : i Λ } of subsets of a topological space X is said to be locally finite (nbd-finite) if each point x in X has an open neighbourhood having non-empty intersection with at most finitely many members of . Example 1.17. Let X = ℝ ∪ {0} , where ℝ is the set of all positive real numbers.Let X have the usal www.ijres.org 11 | Page Some Examples and Counterexamples in Advanced Compactness of Topology topology .Let An = { x X : x≥ n} for each non-negative integer n . Consider collection = { An :n=0,1,2,…}.Then is a locally- finite family in X. Definitin1.18. Let be a covering of a topological space X. A collection is called a refinement of if covers X and each member of is contained in some member of . Definition1.19. A topological space X is said to be Paracompact if every open covering of X has a locally finite refinement which is also an open covering of X. Example 1.20. 1.Every compact space is paracompact .For it let be an open cover of a compact space X, then contains a finite open subcover for X, that is has a finite open refinement , which covers X ⇒ has a locally finite refinement wcich covers X. 2.Every metric space is para compact space.Proof is given by A.H.Stone see in [9]. 3.Any infinite set carrying the particular point topology is not paracompact (see [10] for the definition of particular point topology) , in this case there is a open cover which has no nbd finite refinement. 4.The sorgenfrey lines is paracompact ,but not compact neither locally compact. 5.Every discrete space is paracompact as for it ,every open cover of it has a nbd-finite open refinement. 6.The real line ℝ is paracompact,but not compact.For it by A.H Stone [9], every metrizable space is paracompact. Definition1.21. A topological space X is said to be Lindel¨of if and only if every open cover of X X has a countable sub-cover. Example 1.22. Every compact space is Lindel¨of . For it if X is compact ,then every open cover has finite subcover i.e has countable sub-cover. Theorem1.23. Let (X, d) be a metric space.Then the following statements are equivalent 1.(X, d) compact. 2.(X, d) is countably compact. 3.(X, d) has the Bolzano-Weierstrass property. 4.(X, d) is sequentially compact. The proof is obvious ,see[10] Theorem1.24. For a topological spaces , compactness ⇒ lindelof sequentially compact ⇒ countabe compact compactness ⇒ countably compact. Proof is given see in [2]. II. COUNTEREXAMPLES 2.1.A non compact subset ℝ of in which open cover admits finite subcover. Let (0, 1) as subset of a ℝ and take Cn := (0, - ) ∪ ( , 1) is open cover of (0, 1) such that (0,1) ∪ . Now also we can see that (0,1) ⊂ ∪ ∪ ∪ .
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