Question

Suppose that f : X → Y is a continuous and surjective map between two topological spaces. Determine if the following statements are true or false. If true, prove the statement, if false, give a counter-example.

(a) If X is path-connected, then so is Y.

(b) If X is locally compact, then so is Y.

(c) If X is Hausdorff, then so is Y.

Answer 1

o P & Let y Y EY be two different points in Y Let fly) = y, & fr. ) = y. As f is suspective, such a , M, EX will exist. As X is path connected, choose a Continuous function P: [0, 1] → x sit. p(0) = x & p(l) = n. Then, f (0) f(p (0)) = flr) = y, top (1) = f (p (1)) = f(x) = yz So, fop : [0, 1] → Y is continuous function with top (0) = y, & fop (1) = 4₂ Thus, y is path-connected. b) Let X = (Q, Trisoneta) A Y = (Q, Istandard) se X is space Q with discrete topology Y is space @ with induced topology as subspace of R. Let id: X →y be identity map. Clearly A

open 4 X is locally compact as singleton {n} is & compact subspace of X. But, f(x) = Y is not locally compact as it hos sequence of points converging to irrational point which doesn't lie in Y. So any open interval can't lie in a compact subspace as closure of open interval must be compact if it lies in compact subspace, but then sequence of points in the interval converging to irrational point & hence it can't be sequentially compact. - Let X= R {og & Y = {a, b} with trivial topology. sets in Y are ofy. Let f(x) = re (-0,0) x € (0, 2) Clearly, f is continuous, X is Hansdorff, but f(X) = y is not Hausdooff. we have ie, only open i 0} b ;