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?