example of a topological space which is Normal, locally contractible, semi locally simply connected but not a CW complex.

We know that if a topological space $$X$$ is a CW complex then it is normal, locally path-connected,semi-locally simply connected. And using these properties we can conclude that a space is not a CW complex. But my question is :

Is there a topological space which is normal, locally path connected, semi-locally simply connected, but fails to be a CW complex?

• Take infinite dimensional Hilbert space. – Moishe Kohan Mar 6 at 12:59

If you do not insist on local contractibility and accept examples which are locally path-connected and semi-locally simply connected, you can take the two-dimensional Hawaiian earring $$H = \bigcup_{n=1}^\infty S^2_n$$ where $$S^2_n = \{ x \in \mathbb{R}^3 \mid \lVert x - (1/n,0,0) \rVert = 1/n\}$$. This is a finite-dimensional compact space.