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A colleague in algebra asked me this, and I couldn't answer it. On the Wikipedia page for "epimorphism" it is claimed that in the category of Hausdorff spaces and continuous maps, a function is epi if and only if it has dense range. The "if" case is easy, but I couldn't justify the "only if" case.

This boils down to: let Y be a Hausdorff space, and let X in Y be a closed subset not equal to Y, and not empty. Can you find a Hausdorff space Z and functions f,g:Y->Z such that f and g agree on X, but are not equal. I think, by using a quotient argument, you can assume that X is just a point.

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# Functions separting points in Hausdorff spaces

A colleague in algebra asked me this, and I couldn't answer it. On the Wikipedia page for "epimorphism" it is claimed that in the category of Hausdorff spaces and continuous maps, a function is epi if and only if it has dense range. The "if" case is easy, but I couldn't justify the "only if" case.

This boils down to: let Y be a Hausdorff space, and let X in Y be a closed subset not equal to Y, and not empty. Can you find a Hausdorff space Z and functions f,g:Y->Z such that f and g agree on X, but are not equal. I think, by using a quotient argument, you can assume that X is just a point.