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- (See Exercise 26) Let A be an infinite set, and let H be the set of all fS(A) such that f(x)=x for all but a finite number of elements x of A. Prove that H is a subgroup of S(A).If A is a compact subset of a metric space (X, d) and B is a closed subset of A, prove that B is also compact.If E is a subset of a metric space (X, d), show that E is nowhere-dense in X if and only if E c is dense in X.
- A subset I of a metric space R with the usual metric is compact if and if only it is an interval True FalseLet (X,d) be a metric space , x ϵ X and A ⊑ X be a nonempy set. Prove that d (x ,A) = 0 if and only if every neighborhood of x contains a point of A.Let (X, d) be a metric space with X being an infinite countable set. Show that X is not connected.
- It is given a metric space M with metric d, prove that any epsilon ball is an open set.Let M1 and M2 be two metric spaces and let A is a subset of M1. If f: M1→M2 is continuous, show that f|A: A→M2 is also continuousProve that a closed set in the metric space (S, d) either is nowhere dense in S or else contains some nonempty open set.