Theorem 2

Let X be a topological space, Y a uniform space. For each compact subset K of X and each open subset U of Y, let T(K, U) denote the set of all continuous mappings $u : X \to Y$ such that $u(K) \subset U$. Then the sets $T(K, U)$ generate the topology of compact convergence on $C(X; Y)$.

Let $Y'$ be the Hausdorff uniform space associated with Y (Chapter II, § 3, no. 8) and let $i : Y \to Y'$ be the canonical mapping of Y onto $Y'$. The topology of compact convergence is the coarsest topology for which the mappings $u \to (i \circ u)|K$ of $C(X; Y)$ into $C_u(K; Y')$ are continuous, as K runs through the set of all compact subsets of X ($§ 1,$ no. 4, Proposition 4). Hence we obtain a subbase of the topology of $C_c(X; Y)$ by taking a subbase of the topology of $C_c(K; Y')$ for each compact subset K of X and then taking the union [in $\mathfrak{B}(C(X; Y))$] of the inverse images of these subbases in $C(X, Y)$. On the other hand, every open subset of Y is of the form $\overline{i}^{-1}(U')$, where $U'$ is open in $Y'$ (Chapter II, § 3, no. 7, Proposition 12); hence, for each compact subset $K' \supset K$, $T(K, \overline{i}^{-1}(U'))$ is the inverse image of $T(K, U')$ under the mapping

It thus remains for us to prove the theorem when X is compact and Y is Hausdorff; we shall make these assumptions from now on.

Let us first show that $T(K, U)$ is open in $C_c(X; Y)$. Let $u_0$ be a point of this set; since $u_0(K)$ is compact (Chapter I, § 9, no. 4, Theorem 2, Corollary 1) and contained in the open set $U$, there exists a symmetric entourage $V$ of $Y$ such that $V(u_0(K)) \subset U$ (Chapter II, § 4, no. 3, Proposition 4, Corollary). Let $W$ be the neighbourhood of $u_0$ in $C_c(X; Y)$ consisting of all continuous mappings $u : X \to Y$ such that $(u(x), u_0(x)) \in V$ for all $x \in K$. For such mappings we clearly have $u(K) \subset V(u_0(K)) \subset U$; hence $u \in T(K, U)$ and therefore $W \subset T(K, U)$, which proves our assertion.

Conversely, if $W$ is a neighbourhood of a point $u_0 \in C_c(X; Y)$, let us show that $W$ contains the intersection of a finite number of neighbourhoods of the form $T(K, U)$. We may suppose that $W$ is the set of all $u \in C(X; Y)$ such that $(u(x), u_0(x)) \in V$ for all $x \in X$, $V$ being a given entourage of $Y$. Since $u_0$ is continuous on $X$, it is uniformly continuous (Chapter II, § 4, no. 1, Theorem 2). Let $V_1$ be a symmetric entourage of $Y$, open in $Y \times Y$ and such that $V_1^2 \subset V$. $X$ can be covered by a finite number of compact sets $K_i$ ($1 \leq i \leq n$) such that each $u_0(K_i)$ is $V_1$-small ($1 \leq i \leq n$). Let $U_i$ be the open set $V_1(u_0(K_i))$, and let $u : X \to Y$ be a continuous mapping contained in the intersection of the $n$ sets $T(K_i, U_i)$ (which are neighbourhoods of $u_0$). Then, for every $x \in K_i$, $u(x)$ belongs to $U_i$ and therefore $u_0(x)$ and $u(x)$ are $V_1^2$-close, hence $V$-close. Since each $x \in X$ belongs to some $K_i$, we have $u \in W$ and the proof is complete.

This result leads us to make the following definition: