Open,Closed&amp;Compact

# Open,Closed&amp;Compact - SUMMARY OF PROOFS OF OPENNESS...

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SUMMARY OF PROOFS OF OPENNESS, CLOSEDNESS AND COMPACTNESS MING LI Consider the metric space ( X, d ). Definition 1: A set A X is said to be open , if for any x A , there exists r > 0, such that the open ball B ( x, r ) A . Definition 2-1: A set A X is said to be closed , if A c is open. Definition 2-2: A set A X is said to be closed , if it contains all its limit points. Theorem 1: Definitions 2-1 and 2-2 are equivalent. Definition 3: A set A X is said to be bounded , if there exists R > 0, such that A B (0 , R ). Definition 4-1: A set A X is said to be compact , if every open cover of A has a finite subcover. Definition 4-2: A set A X is said to be compact , if every infinite subset of A has a limit point in A . Definition 4-3: Definition 4’: A set A X is said to be compact ,if every sequence in A has a convergent subsequence with its limit in A . Theorem 2: Definitions (4-1), (4-2) and (4-3) are equivalent. Theorem 3: A compact set is closed. Theorem 4 (Heine-Borel): Suppose A is a subset of the Euclidean space R k . Then A is compact if and only if A is closed and bounded. Theorem 5: Closed subsets of compact sets are compact. Remarks: 1. All definitions above depend on both the space and the metric. For example in the definition of openness, open ball B ( x, r ) is defined as B ( x, r ) = { y X : d ( x, y ) < r } . Limit points must be in the space X . 2. All theorems above apply to any metric space except Theorem 4, which is only true for R k . Now I consider the following examples from our homework problems, and I will use them to illustrate how to prove or disprove that sets are open, closed and compact. In all examples, A

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