CW implies normal

This article gives the statement and possibly, proof, of an implication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property must also satisfy the second topological space property
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This article involves a proof using cellular induction, viz, it inductive construction on the $n$ -skeleton of a cellular space

Statement

Every CW-space (viz every space which can be given a CW-complex structure) is normal, viz it is Hausdorff and any two disjoint closed sets can be separated by disjoint open sets.

Proof

Goal of the proof

Let $X$ be a CW-space. Equip $X$ with a CW-complex structure and let $X^{n}$ denote the $n$ -skeleton with respect to that structure.

Let $A, B \subset X$ be closed subsets. The goal is to construct a function $f : X \to [0, 1]$ such that $f (A) = 0$ and $f (B) = 1$ (this will show that $A$ and $B$ are separated by disjoint open sets).

To construct this $f$ , we construct a family of functions $f_{n}$ on the $n$ -skeletons, such that $f_{n}$ restricted to $X^{m}$ is $f_{m}$ for $m \leq n$ , and such that $f_{n} (A \cap X^{n}) = 0$ and $f_{n} (B \cap X^{n}) = 1$ .

Proof details

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