Please Help Me to Answer all the Parts (a)(b)(c) of the Question, Thank You !!!

Question

Please Help Me to Answer all the Parts (a)(b)(c) of the Question, Thank You !!!

(a) Let X and Y be metric spaces. If X is totally bounded and f: X rightarrow Y is continuous, must f(X) be totally bounded? Prove or provide a counterexample. (b) Let X and Y be metric spaces. If X is totally bounded and f: X rightarrow Y is uniformly continuous, must f(X) be totally bounded? Prove or provide a counterexample. (c) Let X and Y be metric spaces. If X is complete and totally bounded and f: X rightarrow Y is continuous, must f(X) be totally bounded? Prove or provide a counterexample.

Explanation / Answer

a) Consider the following map f: (-1, 1) to real numbers such that f(x) = tan(pi/2) x. Range of f is not bounded.

c) the same counterexample works.

b) Consider the closure of X, as X is totally bounded completion of closure must be compact. Now image of compact is compact. Compact is totally bounded.

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