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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?

Eric Wofsey
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    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.) – Daniel Gerigk Nov 11 '15 at 03:48
  • 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. – ChesterX Nov 11 '15 at 05:07
  • @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. – Daniel Gerigk Nov 11 '15 at 05:53
  • @DanielGerigk, thanks for pointing this out! – ChesterX Nov 11 '15 at 06:16

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