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Let $X$ be a $G$-space and an ordinary CW-complex. We say that $G$ acts cellularly on $X$ if the following holds:

1) For each $g \in G $ and each open cell $E$ of $X$, the left translation $gE$ is again open cell of $X$.

2) If $gE = E$, then the induced map $E \to E$, $x \to gx$ is the identity .

I have a confusion in this definition: Is $X$ necessarily a $G$-space? Or is $X$ only a CW complex?

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    There's no reason ever to talk about actions which are not continuous. If you want to consider an action of some topological group $G$ which is not continuous, then just give $G$ the discrete topology.2012-10-11

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