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Strøm's model category

Arne Strøm? has proven that the category Top of all topological spaces has a structure of a Quillen model category where fibrations are Hurewicz fibrations, cofibrations are closed Hurewicz cofibrations and the role of weak equivalences is played by (strong) homotopy equivalences. The theorem might have been a folklore at the time, but the actual paper has a number of subtleties.

Strøm’s proofs are not that well-known today and use techniques better known to the topologists of that time, and there is consequently a slight controversy among topologists now. One of these is that there are modern reproofs, but these modern techniques essentially use the compactly generated Hausdorff spaces, while Strøm’s proofs succeeded in avoiding that assumption.

The main article is

• Arne Strøm, The homotopy category is a homotopy category, Archiv der Mathematik 23 (1972)

but it depends on earlier results of several authors and mostly his own earlier papers

• Arne Strøm, Note on cofibrations, Math. Scand. 19 1966 11–14 file MR0211403 (35 #2284); Note on cofibrations II, Math. Scand. 22 1968 130–142 (1969) file MR0243525 (39 #4846)

One modern re-proof can be found in

• May and Sigurdsson, Parametrized homotopy theory