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Is there a classification of those (full) subcategories of the category $\mathbf{Top}$ of topological spaces that are closed under arbitrary products and arbitrary coproducts in $\mathbf{Top}$?

EDIT: I’m especially interested in the case where such a subcategory contains all CW complexes.

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    $\begingroup$ There is a result of Kannan saying that if we require closedness under products, coproducts, subspaces and quotient spaces, then there are only trivial cases. (The paper is called Reflexive cum coreflexive subcategories on topology.) Some results are known about combinations of 2 or 3 operations from this list. (In particular, products and subspaces = epireflective subcategories, coproducts and quotients = coreflective subcategories.) But I do not know whether classes closed under products and coproducts were studied. $\endgroup$ – Martin Sleziak Dec 9 '14 at 14:31
  • $\begingroup$ Thank you for the reference, I'll have a look. I'm also going to narrow down the scope of the question a bit. $\endgroup$ – johndoe Dec 9 '14 at 14:47
  • $\begingroup$ The first example that comes to mind is the category consisting of those spaces that can be constructed from $CW$-complexes using products and coproducts, together with all non-Hausdorff spaces with cardinality strictly greater than the continuum, infinitely many non-compact path components having the Monster finite simple group as a direct factor of their fundamental groups, at least 17 path components having the Hawaiian earring as a retract, and allowing a faithful action of a Tarski monster by homeomorphisms. It seems unlikely that there's a reasonable classification. $\endgroup$ – Jeremy Rickard Dec 13 '14 at 11:21
  • $\begingroup$ @JeremyRickard Can you explain the numerology behind your example, or is it a joke? $\endgroup$ – johndoe Dec 13 '14 at 11:28
  • $\begingroup$ @johndoe I was just trying to make the point that you can come up with almost arbitrarily complicated examples of subcategories that are closed under products and coproducts, and that therefore it's hard to see what a reasonable classification would look like. $\endgroup$ – Jeremy Rickard Dec 13 '14 at 11:30

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