Definition 53.2.1. Let $G$ be a topological group. A *$G$-set*, sometime called a *discrete $G$-set*, is a set $X$ endowed with a left action $a : G \times X \to X$ such that $a$ is continuous when $X$ is given the discrete topology and $G \times X$ the product topology. A *morphism of $G$-sets* $f : X \to Y$ is simply any $G$-equivariant map from $X$ to $Y$. The category of $G$-sets is denoted *$G\textit{-Sets}$*.

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