Characterizations of Compactness for Metric Spaces
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چکیده
Definition. Let X be a metric space with metric d. (a) A collection {G α } α∈A of open sets is called an open cover of X if every x ∈ X belongs to at least one of the G α , α ∈ A. An open cover is finite if the index set A is finite. (b) X is compact if every open cover of X contains a finite subcover. Definition. Let X be a metric space with metric d and let A ⊂ X. We say that A is a compact subset if the metric space A with the inherited metric d is compact. Examples: Any finite metric space is compact. As an exercise show directly from the definition that the subset K of R consisting of 0 and the numbers 1/n, n = 1, 2,. .. is compact. Definition. A subset A of X is relatively compact if the closure A ⊂ X is a compact subset of X. Definition. A metric space is called sequentially compact if every sequence in X has a convergent subsequence. Definition. A metric space is called totally bounded if for every ǫ > 0 there is a finite cover of X consisting of balls of radius ǫ. THEOREM. Let X be a metric space, with metric d. Then the following properties are equivalent (i.e. each statement implies the others): (i) X is compact. (ii) X has the Bolzano-Weierstrass property, namely that every infinite set has an accumulation point. * (iii) X is sequentially compact, i.e. every sequence has a convergent subsequence. (iv) X is totally bounded and complete. * If A is a subset of X then p is called an accumulation point if every neighborhood of p contains a point q ∈ A so that q = p. In Rudin's book the terminology 'limit point' is used for this.
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