continuous composition for locally compact spaces on the compact open topology

Let \((X, \mathcal{T}_X)\), \((Y, \mathcal{T}_Y)\) and \((Z, \mathcal{T}_Z)\) be topological spaces, such that \(Y\) is locally pseudoregular compact

Then the composition

\begin{align*} \sigma: \mathrm{Map}(Y,Z) \times \mathrm{Map}(X,Y) \rightarrow \mathrm{Map}(X,Z) \end{align*}

is continuous with respect to the product topology and the compact open topology of the mapping spaces

to show

\begin{align*} \sigma^{-1}( \mathcal{O}_{K,U}) \subseteq \mathrm{Map}(Y,Z) \times \mathrm{Map}(X,Y) \end{align*}

suppose \((g,f) \in \sigma^{-1}[\mathcal{O}_{K,U}]\) Then \(g \circ f \in \mathcal{O}_{K,U}\), hence \(f[K] \subseteq g^{-1}[U]\)

Since \(Y\) is locally compact, for all \(y \in F[K] \subseteq g^{-1}[U]\), there exists a compact neighbourhood \(g \in K_y \subseteq g^{-1}[U]\), such

Hence

\begin{align*} f[K] \subseteq& \bigcup_{y \in f[K]} U_y \subseteq g^{-1}[U] \end{align*}

By compactness, there exists a finite covering \(U_1,...,U_n\) of \(f[K]\) Let \(L = K_1 \cup ... \cup K_n\) be compact subsets of \(Y_i\), then for

\begin{align*} \mathcal{O}_{L,U} \times \mathcal{O}_{K,V} \subseteq \sigma^{-1}[\mathcal{O}_{K,U}] \end{align*}

continuous map and subbasis

Let \(\mathcal{O}_{K,U}\) be open with \(K \subeteq X\) compact and \(U \subseteq Z\) open.

Then for \((g,f) \in \sigma^{-1}[\mathcal{O}_{K,U}]\), it follows by preimage and subsets

\begin{align*} g \circ f[K] \subseteq& U && \vert g^{-1}[-]\\ f[K] \subseteq g^{-1} \circ g \circ f[K] \subseteq& g^{-1}[U] \end{align*}

Since \(Y\) is locally pseudoregular compact, there exists an open cover of \(g^{-1}[U]\) with compact supersets (cf. locally pseudoregular compact and open set as union of open sets with compact supersets). Because \(K \subseteq g^{-1}[U]\) is compact, there exists a finite covering \(O_1,...,O_n\) with compact supersets \(C_i \supseteq O_i\).

Let \(L = \bigcup_{i=1}^n C_i\) be the finite union of those compact sets, hence also compact (cf. finite union of compact sets) and \(V \coloneqq \bigcup_{i=1}^n O_i\).

Then

\begin{align*} (g,f) \in \mathcal{O}_{L,U} \times \mathcal{O}_{K,V} \end{align*}

and furthermore since \(V \subseteq L\), it follows, that

\begin{align*} \mathcal{O}_{L,U} \times \mathcal{O}_{K,V} \subseteq \sigma^{-1}[ \mathcal{O}_{K,U}] \end{align*}

hence \(\sigma^{-1}[\mathcal{O}_{K,U}]\) is a neighbourhood of all points, hence open (cf. openness and neighbourhood for all points)