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I've recently started stydying algebraic topology and am now learning about cell complexes. I understand the iterative construction of such spaces, but I lack any intuition concerning the limitations on the possible resulting structures.

I thus have two questions for which I am seeking, preferably, non-highly technical answers:

1) What types of structures can, and cannot, be created iteratively from cells?

2) What types of spaces are, and are not, homotopic to the spaces created from cells?

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    $\begingroup$ Regarding your second question, it can be said that every cell complex is homotopy equivalent to a CW complex. (I don't know whether there is a space that has the homotopy type of a CW complex but is not a cell complex.) $\endgroup$ – Daniel Gerigk Nov 11 '15 at 3:48
  • $\begingroup$ I can give you a non-example : Hawaiian earrings is not of homotopy type of a CW complex. The reason is that every CW-complex is necessarily locally contractible, which fails for the earring. $\endgroup$ – ChesterX Nov 11 '15 at 5:07
  • $\begingroup$ @ChesterX Indeed, the Hawaiian earring is not of the homotopy type of a CW complex, but not because it is not locally contractible! The property of being locally contractible is not homotopy invariant, as explained by studiosus here. $\endgroup$ – Daniel Gerigk Nov 11 '15 at 5:53
  • $\begingroup$ @DanielGerigk, thanks for pointing this out! $\endgroup$ – ChesterX Nov 11 '15 at 6:16

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