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

$\text{Map}(E, E) \times E \longrightarrow E,\quad (f, e) \longmapsto f(e)$

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

$\text{Map}(E, E) \times \text{Map}(E, E) \to \text{Map}(E, E)$

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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