# Kerodon

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Warning 2.4.2.18. 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 2.4.2.16 coincides with the simplicial enrichment of Example 2.4.1.5. 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 2.1.7.8).

• 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 .