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It’s well know that finite CW complexes are not closed under loop spaces (viewed as homotopy pullbacks). Just consider the circle. Are countable-dimensional CW complexes with countably many cells at each level closed under these homotopy pullbacks? I can’t think of a counter-example but am unsure if it’s true in general.

CuriousKid7
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    I think so and I think the argument should go like this: show that up to homotopy these are precisely the CW complexes with $\pi_i$ at most countable for each $i$ and each basepoint. Then this class of CW complexes is obviously closed under taking loop spaces. – Qiaochu Yuan Jun 27 '22 at 17:57
  • @QiaochuYuan Got it, thanks. This is Theorem 6.1 (pg. 137) of The Topology of CW Complexes by Lundell and Weingram. – CuriousKid7 Jun 27 '22 at 23:47
  • It is true and some details are found in my answer here (note that every countable CW complex is homotopy equivalent to a countable locally finite CW complex which is an ANR). – Tyrone Jun 28 '22 at 11:16

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