Let $(X, \mathcal{E})$ be a CW complex and $\mathcal{E'} \subseteq \mathcal{E}$ be a subset of cells such that $\mathcal{E'}$ is a CW decomposition of the space $X' = \bigcup_{e \in \mathcal{E'}} e \subseteq X$. Is $\mathcal{E'}$ then necessarily a subcomplex of $\mathcal{E}$?
I know that if $\mathcal{E'}$ is finite, then $X'$ is compact and thus closed in $X$, so that $\mathcal{E'}$ is a subcomplex of $\mathcal{E}$. But can we say anything in the infinite case? Intuition tells me there should be a counterexample, but I'm not able to come up with one.