For every group $G$ there is a semilocally simply-connected cell complex $X$ with $\pi_1(X)=G$

Question

Proposition For every group $G$ there is a semilocally simply-connected cell complex $X$ with $\pi_1(X)=G$.

I know that for every group $G$ there is a cell complex $X$ with $\pi_1(X)=G$. The construction of such $X$ is as follows :

(This is from Hatcher, Algebraic Topology, Corollary 1.28) Choose a presentation $<g_\alpha ~|~r_\beta >$ of $G$. Then let $X$ be the space obtained by attaching 2-cells $e^2 _\beta$ to the wedge sum $\bigvee _\alpha S^1 _\alpha$ by the loops specified by the words $r_\beta$.

Intuitively, I think that this cell complex $X$ is in fact semilocally simply-connected, but I cannot show this. How do I have to proceed?

Answer 1

This community wiki solution is intended to clear the question from the unanswered queue.

Lee Mosher is right in his comment. See Hatcher Proposition A.4:

Each point in a CW complex has arbitrarily small contractible open neighborhoods, so CW complexes are locally contractible.

Note that Hatcher uses "cell complex" and "CW complex" as synonyms (see page 5).