Editing CW complex (section)

=== Non CW-complexes ===

* An infinite-dimensional [[Hilbert space]] is not a CW complex: it is a [[Baire space]] and therefore cannot be written as a countable union of ''n''-skeletons, each of which being a closed set with empty interior. This argument extends to many other infinite-dimensional spaces.
* The [[hedgehog space]] <math>\{re^{2\pi i \theta} : 0 \leq r \leq 1, \theta \in \mathbb Q\} \subseteq \mathbb C</math> is homotopy equivalent to a CW complex (the point) but it does not admit a CW decomposition, since it is not [[Contractible space#Locally contractible spaces|locally contractible]].
* The [[Hawaiian earring]] has no CW decomposition, because it is not locally contractible at origin. It is also not homotopy equivalent to a CW complex, because it has no good open cover.