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Consider the definition of CW complex from wikipedia. It is assumed that the space is Hausdorff.

Are there problems if we drop this assumption? What is an example of a space satisfying all the CW complex axioms except this condition?

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    For one thing, it is really nice to have one's compact sets be closed!2011-10-05
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    The right way to think about CW complexes would rather be the inductive definition, which is also mentioned in the wikipedia article. The spaces constructed in this way are Hausdorff.2012-01-03

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