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A topological group is a group $G$ together with a topology on the elements of $G$ such that the group operation and group inverse function are both continuous (with respect to the topology).

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Defining the classifying space of a group acting on a set

As in Tom Goodwillie's comment, if you take the construction that you discuss for a $G$-set $X$, the equivariant homotopy type of the space obtained before you quotient out by the action of $G$ is cal …
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