Example 5.30.2. Let $E$ be a set. We can endow the set of self maps $\text{Map}(E, E)$ with the compact open topology, i.e., the topology such that given $f : E \to E$ a fundamental system of neighbourhoods of $f$ is given by the sets $U_ S(f) = \{ f' : E \to E \mid f'|_ S = f|_ S\} $ where $S \subset E$ is finite. With this topology the action

is continuous when $E$ is given the discrete topology. If $X$ is a topological space and $X \times E \to E$ is a continuous map, then the map $X \to \text{Map}(E, E)$ is continuous. In other words, the compact open topology is the coarsest topology such that the “action” map displayed above is continuous. The composition

is continuous as well (as is easily verified using the description of neighbourhoods above). Finally, if $\text{Aut}(E) \subset \text{Map}(E, E)$ is the subset of invertible maps, then the inverse $i : \text{Aut}(E) \to \text{Aut}(E)$, $f \mapsto f^{-1}$ is continuous too. Namely, say $S \subset E$ is finite, then $i^{-1}(U_ S(f^{-1})) = U_{f^{-1}(S)}(f)$. Hence $\text{Aut}(E)$ is a topological group as in Definition 5.30.1.

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