CW implies locally contractible

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., locally contractible 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 a CW-complex structure) is locally contractible.

Proof

This statement follows from the following result:

Given a CW-complex and an open subset containing a CW-subcomplex, there exists an smaller open set containing the subcomplex, for which the subcomplex is a strong deformation retract

The implication is not immediate, because the point that we start with may not be a $0$-cell.

References

• Lundell and Weingram, P. 63