CW implies perfectly 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 (i.e., CW-space) must also satisfy the second topological space property (i.e., perfectly normal space)
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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

Any CW-space (viz a space that can be given the structure of a CW-complex) is perfectly normal, viz it is normal and every closed subset is a G-delta subset.

References

• Topology of CW-complexes by A. T. Lundell and S. Weingram, P. 54