I have a problem with reading the proof of the proposition A.1 in Hatcher's Algebraic Topology book.
In the last sentence he says $A \cup e^n_\alpha$ is a finite subcomplex containing $e^n_\alpha$. To say like this, we should make sure that the closure $\overline{ e^n_\alpha }$ is a subset of $A \cup e^n_\alpha$. How can I do this? My idea was using that $X$ is Hausdorff, but this does not seem to be the same logic to Hatcher's because he states this Hausdorff property two pages later.