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Let $(X, \rho_{X})$ and $(Y, \rho_{Y})$ be two compact metric spaces. Consider the metric space $(M_{XY}, \rho)$, where $M_{XY}$ is the set of any mappings from X to Y and $\rho(f,g) := \sup_{x \in X}\{\rho_{Y}(f(x), g(x))\}$. Suppose $C_{XY}$ is the set of continuous mappings from X to Y. Show that $C_{XY}$ is closed in $M_{XY}$.

I have shown that $(M_{XY}, \rho)$ is indeed a well defined complete metric space. To show the closedness of $C_{XY}$, I suspect that I should use the generalised Arzela-Ascoli theorem stating that "Given $(X, \rho_{X})$ and $(Y, \rho_{Y})$ be two compact metric spaces, and $D = \{f_{\alpha}\}_{\alpha \in I} \subset C_{XY}$ be a set of continuous mappings from $X$ to $Y$. Then $D$ is relatively sequentially compact if and only if $D$ is equicontinuous." But I really don't know how to show that $C_{XY}$ is equicontinuous.

Appreciate any help. Thanks.

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We can show that the complement of $C_{XY}$ is open: if $f$ belongs to this set, then there is some $x_0$ such that $f$ is not continuous at $x_0$. Hence, for some positive $\delta$ and some sequence $(x_n)_{n\geqslant 1}$ converging to $x$, we have $$\rho_Y(f(x_n),f(x_0))\gt 3\delta.$$ We can show that each function $g$ such that $\rho(f,g)\lt\delta$ is discontinuous at $x_0$. Indeed, since $$3\delta\lt \rho_Y(f(x_n),f(x_0))\leqslant \rho_Y(f(x_n),g(x_n))+\rho_Y(g(x_n),g(x_0))+\rho_Y(g(x_0),f(x_0))\leqslant 2\rho(f,g)+\rho_Y(g(x_n),g(x_0)),$$ we get $\rho_Y(g(x_n),g(x_0))\gt \delta$ for each $n$.

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  • $\begingroup$ That's elegant. Thank you very much. $\endgroup$ – theflyingwolves Nov 9 '14 at 13:22

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