Talk:Open and closed maps

Sufficient for continuity?

Regarding "...a function f : X → Y is continuous...if the preimage of every closed set of Y is closed in X." at the end of the second paragraph:

Forgive me if I'm mistaken, but in general, preimages of closed sets being closed does not ensure continuity. E.g. for f: [0,1] -> (0,1] where f(x) = x (for x != 0) and f(0) = 1/2, f is not continuous. Perhaps something is assumed that I missed? 203.150.100.189 08:38, 20 March 2007 (UTC)

Please read again: "if the preimage of every open set of Y is open in X". For your example, the preimage of ]1/4,3/4[ is then {0}U]1/4,3/4[ which is not open.--133.11.80.84 07:05, 3 September 2007 (UTC)

203: Pass to complements. Suppose f is continuous and take a closed set V. Then the complement of V, which I'll denote C(V), is open, so it pulls back to an open set. But the preimage of V is the complement of the preimage of C(V), so it's closed. The other way: Suppose f pulls back closed sets to closed sets. A similar argument gives you that f pulls back open sets to open sets, so f is continuous. HTH. Druiffic (talk) 04:26, 15 November 2008 (UTC)Druiffic

Yeah, but Druiffic, 203's example proves you wrong! consider the closed set V=[1/8,1/4] u [3/4,1] Then C(V) = (0,1/8) u (1/4,3/4) is clearly open. Next, f^-1(C(V))= (0,1/8) u (1/4,3/4) u {0} = [0,1/8) u (1/4,3/4) is clearly NOT open! Thus, although f pulls back closed sets just fine, this f is not continuous and clearly fails to pull back open complements into open complements.67.198.37.16 (talk) 17:01, 30 June 2016 (UTC)

Silly me -- I am wrong. In the target space, the interval (0,1/8) is NOT open -- it is half-closed. That is, in the target space, the interval (0,1/8] is fully closed, and it does not pull back to a closed set. This is a subtle point: the definition of a closed set in the target space is NOT the same as the definition of a closed set in the natural topology on the reals -- in particular, the entire target space is considered to be closed. Its an interesting, illuminating failed counter-example, illustrating that one mst be careful to use the definition of "open" and "closed" correctly. 67.198.37.16 (talk) 18:17, 30 June 2016 (UTC)

Surjective claim

Unless I'm going crazy, "An open map is also closed if and only if it is surjective" is definitely not true. Just include X into two (disjoint) copies of itself. —Preceding unsigned comment added by 24.19.0.156 (talk) 00:54, 8 February 2011 (UTC)

This statement appears to have been removed from the article with this edit, about a week after you complained. 67.198.37.16 (talk) 18:27, 30 June 2016 (UTC)

Example

I was wondering if isometries in metric spaces are open. — Preceding unsigned comment added by Noix07 (talk • contribs) 15:35, 16 December 2013 (UTC)