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take_home1 - Take—Home Final Problems 1 Let(X p be a...

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Unformatted text preview: Take—Home Final Problems 1} Let (X, p] be a metric space. Show that FLT? y) 1 + pm at) is also a metric on X which induces the same topology. Note that 1' is a bounded metric on X: 113:, y) s: 1 for all or, y E X. Tim: at) = 2} Let (X, p] be a metric space. Show that [a] If {333:1 is a sequence in X. then {3,- :1 has a subsequence converging to 3: if and only it" le inf{p(:cm, film 2 r1} = D. (b) X is compact if and only it any continuous function f : X —!r E is bounded. 3) Let X be a metrizeable topological space. Show that X is compact if and only it" any metric inducing the given topology on X is bounded. ...
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