Does a compact semilocally simply connected geodesic space have the homotopy type of a compact CW complex? Actually what I'd like to know is whether the fundamental group of such a space is finitely presented. 


Edit: As Bruno Martelli notes, this is obviously false, but the question of whether the fundamental group is finitely presented, which is what I really want to know, is still open.



Does a compact semilocally simply connected geodesic space have the homotopy type of a CW complex?
The bouquet of infinitely many shrinking 2-spheres in $\mathbb R^3$ centered in $(0,0,n)$ and of radius $n$ is a compact simply connected geodesic space, which is not homotopic to a compact CW complex.


