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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?

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  • $\begingroup$ Take infinite dimensional Hilbert space. $\endgroup$ – Moishe Kohan Mar 6 at 12:59
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There are plenty of examples.

As Moishe Cohen remarked in his comment, any infinite dimensional Hilbert space will do (or more generally, any infinite dimensional normed linear space, or even more generally, each infinite dimensional convex subset of such a space).

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.

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