CW implies locally contractible

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.