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Is there a non-$T_2$ space $(X,\tau)$ with the following property?

For all topological spaces $A$ and continous maps $f,g:A\to X$ the set $\{a\in A: f(a) = g(a)\} \subseteq A$ is closed.

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1 Answer 1

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Apply this to $A= X \times X$ with $f$ and $g$ the two projection on $X$ and it gives you that the diagonal $\Delta_X \subset X \times X $ is closed, and this is well known to be equivalent to $X$ is $T_2$.

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