As is well known, the homeomorphism group of a compact Hausdorff space is a topological group. The same is true for locally compact locally connected Hausdorff spaces, but it is false in general. Now what if the space is compactly generated and weak Hausdorff and the homeomorphism group carries the compactly generated topology stemming from the CO topology? Is it necessarily a topological group?


I've been thinking about this question recently, and came across this post. Here is a partial answer.

If $X$ is a locally compact Hausdorff space, then Arens' 1946 $g$-topology on the group $\mathrm{Homeo}(X)$ is an admissible group topology. Therein, it turns out to be the restriction of Fox's 1945 compact-open topology on $\mathrm{Homeo}(X^+)$ to $\mathrm{Homeo}(X)$, where $X^+$ is the one-point compactification. The $g$-topology is finer than the compact-open topology on $\mathrm{Homeo}(X)$.

Admissible means that the evaluation function $\mathrm{Homeo}(X) \times X \to X ~;~ (f,x) \mapsto f(x)$ is continuous. Group topology on a group $G$ means that the fraction function $G \times G \to G ~;~ (a,b) \mapsto a^{-1} b$ is continuous. If the locally compact space $X$ is not compact, then the compact-open topology (called $k$-topology within Arens' paper) on $\mathrm{Homeo}(X)$ is an admissible monoid topology, but inversion is not guaranteed to be continuous. This is remedied by Arens' $g$-topology refining it.

Now, suppose more generally that $X$ is a compactly generated Hausdorff space, as surveyed in Steenrod's 1967 paper, "A convenient category of topological spaces." Endow $\mathrm{Homeo}(X)$ with the compactly generated topology induced from the compact-open topology. It follows from methods of that paper, especially from Section 5, that $\mathrm{Homeo}(X)$ becomes a group object in the category $CGHaus$ of compactly generated Hausdorff spaces, which is a retract of the category $Haus$ of Hausdorff spaces. However, they possess different products. So I don't know if the topology you specified makes $\mathrm{Homeo}(X)$ into a group object in $Haus$.

Culturally speaking, $CGHaus$ is preferred by most algebraic topologists, whereas $Haus$ or $Top$ is preferred by most general topologists.

In categorical language, a topological group ends up being a group object in the category $Top$ of all topological spaces and continuous functions, just like an algebraic group is a group object in the category of all algebraic varieties (over a given field) and regular functions.


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