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June 27

Continuous on Closed Cover

What properties does a topological space X need to satisfy so that if f: X -> X is continuous on each set in a closed cover of X, f is continuous on X? This is not a homework question, I just can't remember and, for some reason, can't seem to figure it out at the moment. Thank you for any help :-) 209.252.235.206 (talk) 07:08, 27 June 2012 (UTC)[reply]

I doubt there are any properties that non-trivially imply this. Euclidean space lacks this property, for example, because the collection of singletons form a closed cover, and every function is continuous on a singleton.--2601:9:1300:5B:21C:B3FF:FEC6:8DA8 (talk) 08:55, 27 June 2012 (UTC)[reply]

Wow, I'm getting sloppy these days! Completely overlooked that; how about a closed cover containing 2 sets E and F? I'm thinking more along the lines of cutting an interval in half and showing f is continuous on each part; you can even assume that each is homeomorphic X. 209.252.235.206 (talk) 09:07, 27 June 2012 (UTC)[reply]

I don't think there's a property for spaces that works for every closed cover, but there is a criterion for the closed cover: it needs to be locally finite (every point has a neighborhood that intersect only finitely many closed sets in the cover). Money is tight (talk) 11:18, 27 June 2012 (UTC)[reply]

Thank you:-) — Preceding unsigned comment added by 209.252.235.206 (talk) 04:43, 29 June 2012 (UTC)[reply]

The answer is very simple for T1 spaces X: in this case, the condition is equivalent to: X has the discrete topology. Indeed, if it has the discrete topology, every map on X is continuous and the property trivially holds; conversely, since in a T1 space X points are closed sets, they constitue a closed covering; thus the property implies that every map f: X -> X is continuous. This also implies that any subset of X is closed (there are maps which are constant on it and on its complement). The answer is less simple for general topological spaces, not necessarily T1; one should argue similarly using the covering made by the closures of single points, and obtain a characterization in terms of the structure of the lattice of the closed sets of X. --pm a 10:48, 29 June 2012 (UTC)[reply]