(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let X be a topological space, Y a Hausdorff space, and let f:X -> Y and g:X -> Y be continuous. Show that {x [tex]\in[/tex] X : f(x) = g(x)} is closed. Hence if f(x) = g(x) for all x in a dense subset of X, then f = g.

2. Relevant equations

Y is Hausdorff => for every x, y in Y with x != y, there exist disjoint open sets U, V with x in U and y in V.

f continuous iff f^{-1}(V) is open in X whenever V is open in Y, iff f^{-1}(F) is closed whenever F is closed.

3. The attempt at a solution

I could show the set is closed by proving that its complement is open, but do I want to take that route? The complement is {x in X : f(x) != g(x)}. So is this itself a Hausdorff space? I'm not sure if this is the right way to go with this, or even if it's correct. Thanks, as always, for any help.

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# Homework Help: Topology: Hausdorff Spaces

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