So I wanted to verify if this was true since it doesn't seem to be in Hatcher. If we have a cell complex $X$ and a closed subset $C\subseteq X$. Can we refine the cellular structure of $X$ so that $C$ is a subcomplex of $X$? We can assume $C$ is compact if necessary.
Cell complex and closed subset
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algebraic-topology
cw-complexes
1 Answers
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If this were the case, all compact subsets of $\Bbb R^n$ would be cellular complexes. Which is not true, since some of them are not even semilocally simply connected.