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Warning Let $\operatorname{Top}_{\mathrm{LCH}}$ denote the full subcategory of $\operatorname{Top}$ spanned by the locally compact Hausdorff spaces. Then we can view $\operatorname{Top}_{\mathrm{LCH}}$ as a topologically enriched category, where we endow each of the sets

\[ \operatorname{Hom}_{\operatorname{Top}_{\mathrm{LCH} }}( X, Y) = \{ \text{Continuous functions $f: X \rightarrow Y$} \} \]

with the compact-open topology, generated by open sets of the form $\{ f \in \operatorname{Hom}_{\operatorname{Top}}(X,Y): f(K) \subseteq U \} $ where $K \subseteq X$ is compact and $U \subseteq Y$ is open. On this subcategory, the simplicial enrichment of Example coincides with the simplicial enrichment of Example Beware that some technical issues arise if we allow spaces which are not locally compact:

  • Given topological spaces $X$, $Y$, and $Z$, the composition map

    \[ \operatorname{Hom}_{\operatorname{Top}}(Y,Z) \times \operatorname{Hom}_{\operatorname{Top}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{Top}}(X,Z) \quad \quad (g,f) \mapsto g \circ f \]

    need not be continuous (with respect to the compact-open topologies on $\operatorname{Hom}_{\operatorname{Top}}(X,Y)$, $\operatorname{Hom}_{\operatorname{Top}}(Y,Z)$, and $\operatorname{Hom}_{\operatorname{Top}}(X,Z)$) when $Y \notin \operatorname{Top}_{\mathrm{LCH}}$. Consequently, the construction of compact-open topologies does not determine a topological enrichment of $\operatorname{Top}$ (in the sense of Example

  • Given topological spaces $X$ and $Y$, a function $| \Delta ^ n | \rightarrow \operatorname{Hom}_{\operatorname{Top}}(X,Y)$ which is continuous (for the compact-open topology on $\operatorname{Hom}_{\operatorname{Top}}(X,Y)$) need not correspond to a continuous function $| \Delta ^ n | \times X \rightarrow Y$ when $X \notin \operatorname{Top}_{\mathrm{LCH} }$.

One can remedy these difficulties by replacing $\operatorname{Top}$ by the subcategory of compactly generated weak Hausdorff spaces introduced in [MR0251719].