CW complexes are T$_1$

Question

Given the constructive definition of CW-complexes (i.e. the one Hatcher gives in his Algebraic Topology book) how would one prove that every singleton in closed. He states in page 522 that every point pulls back to closed subsets of the closed discs $D_\alpha^n$ under every characteristic map $\Phi_\alpha$. But I do not see how this is immediate.

Answer 1

Fix a point $x$ in your CW-complex $X$ and a cell $e^n_\alpha$ with attaching map $\varphi_\alpha:\partial D^n\to X^{n-1}$ and characteristic map $\Phi_\alpha:D^n\to X$. By induction on $n$, you can assume the $(n-1)$-skeleton $X^{n-1}$ is $T_1$. If $x\not\in X^{n-1}$, then $\Phi_\alpha^{-1}(\{x\})$ has at most one point and thus is closed (since $\Phi_\alpha$ is injective on the interior of $D^n$ and that is the only part of $D^n$ that it maps outside of $X^{n-1}$). If $x\in X^{n-1}$, then $\{x\}$ is closed in $X^{n-1}$ by the induction hypothesis, so $\Phi_\alpha^{-1}(\{x\})=\varphi_\alpha^{-1}(\{x\})$ is closed in $\partial D^n$ and hence also in $D^n$.