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The problem

Let $X$ be an infinite set with the cofinite (finite complement) topology and let $Y$ be a Hausdorff space. Characterize the continuous functions from $X$ to $Y$.

What I have so far

For a function $f:X \rightarrow Y$ to be continuous, we have that

$f$ is continuous $\iff$ for any open set $U$ of $Y$, $f^{-1}(U)$ is open in $X$,

or, equivalently:

$f$ is continuous $\iff$ for any closed set $B$ of $Y$, $f^{-1}(B)$ is closed in $X$.

The closed sets of $X$ are all the finite subsets of $X$ or all of $X$.

Since every finite point set $W$ in a Hausdorff space is closed, we must have that $f^{-1}(W)$ is either a finite subset of $X$ or all of $X$.

But what about the infinite closed subsets of $Y$? I don't really know where to go from here. Am I on the right track here? Should I be looking at the closed sets at all?

Any help appreciated!

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up vote 5 down vote accepted

Hint: Take $x_1, x_2 \in X$ with $f(x_1) \ne f(x_2)$. As $Y$ is Hausdorff, there are disjoint open neighbourhoods. Now look at their preimages.

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Thanks, I got it now! – Maethor Oct 27 '12 at 12:38

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