If A is a subset of a metric space X, then show that A = (x: each neighbourhood of x intersects A}
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- A subset I of a metric space R with the usual metric is compact if and if only it is an interval True FalseShow that the interval (a,b) in R with the discrete metric space is locally compact but not compact.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 continuous
- Let (X, d) be a complete metric space and let F ⊆ X be a closed set. Show that F is complete.Let (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.It is given a metric space M with metric d, prove that any epsilon ball is an open set.
- 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.Let E and F be compact sets in a metric space (X,d). Show that E U F is compact using the definition of compactness.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.