CW implies locally contractible
From Topospaces
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 skeleton of a cellular space
Statement
Any CWspace (viz, a space that can be given a CWcomplex structure) is locally contractible.
Proof
This statement follows from the following result:
Given a CWcomplex and an open subset containing a CWsubcomplex, 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 cell.
References
 Lundell and Weingram, P. 63