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I can't find a proof of this result, can anyone help me?

Let $X$ be a topological space and $C(X,X)$ the space of all continuous functions from $X$ to itself. Suppose $X$ is regular, then $C(X,X)$, endowed with the compact-open topology, is regular.

Thanks.

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    This follows immediately from the definitions. Let $K \subset X$ be compact and $O \subset Y$ be open in the regular space $Y$. Let $f \in N(K,O)$ then $f(K)$ is compact and contained in the open $O$, so there is $U$ such that $f(K) \subset U \subset \overline{U} \subset O$ by regularity of $Y$. But then $f \in N(K,U) \subset \overline{N(K,U)} \subset N(K,\overline{U}) \subset N(K,O)$, hence $C(X,Y)$ is regular.2011-10-02

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The separation properties, such as $T_0, T_1$, Hausdorff, regular and completely regular- Hausdorff are inherited by $C(X,Y)$ under compact-open topology from its target space ($Y$). This is a standard exercise. See for example Section 46 in Munkres.

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    @t.b. Yes, you are right. Thanks for the correction.2011-10-02