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For a CW-complex, locally compact, metrizable, first countable and locally finite are equivalent conditions. A proof is available in https://epub.ub.uni-muenchen.de/4524/1/4524.pdf. I need the same result for cellular topological spaces, i.e. cellular objects of the Quillen model structure of spaces. It is possible to adapt the arguments of the Fritsch and Piccinini paper but I thought that someone wrote down the proof somewhere that I could cite.

Any bibliographical reference for cellular spaces ? And by the way, is there any textbook expounding the proof for CW-complexes ? This fact is not in Hatcher's book "Algebraic Topology", not in Kozlov's book "Combinatorial Algebraic Topology" and not in Rotman's book "An introduction to Algebraic Topology". I am a bit surprised.

Def: A cellular topological space is a space $X$ such that the map $\varnothing\to X$ is a transfinite composition of pushouts of the generating cofibrations $\mathbf{S}^{n-1}\subset \mathbf{D}^n$ for $n\geq 0$.

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  • $\begingroup$ What would be your reference for a text about this model structure of spaces? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Apr 30, 2021 at 11:15
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    $\begingroup$ The theorem (for CW complexes) is given in Fritsch and Piccinini's book Cellular Structures in Topology (see Proposition 1.5.17, for instance). It also appears in tom Dieck's books (for simplicial complexes). Could you be more specific about what you accept as a 'cellular object'? $\endgroup$
    – Tyrone
    Commented Apr 30, 2021 at 11:56
  • $\begingroup$ @მამუკაჯიბლაძე A reference about the Quillen model structure of spaces ? Mark Hovey's book for example. $\endgroup$
    – Philippe Gaucher
    Commented Apr 30, 2021 at 12:10
  • $\begingroup$ @Tyrone I edit the question. Your book is there epub.ub.uni-muenchen.de/4493/1/4493.pdf. $\endgroup$
    – Philippe Gaucher
    Commented Apr 30, 2021 at 12:19

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