Paths and cylinders
A continuous function is a Hurewicz cofibration if it satisfies the homotopy extension property for all spaces and with respect to the standard notion of left homotopy of topological spaces given by the standard topological interval object/cylinder object.
More generally, one may speak of morphisms in any category with weak equivalences having the homotopy extension property with respect to a chosen cylinder object, one speaks of h-cofibrations.
A continuous function is a Hurewicz cofibration if it satisfies the homotopy extension property in that:
for any topological space ,
all continuous functions , such that
and any left homotopy such that
there is a homotopy such that
A Hurewicz cofibration (def. 1) is called a closed cofibration if the image is a closed subspace in .
If is closed and the inclusion is a cofibration, then the pair is called an NDR-pair.
There is also a version of the definition for pointed spaces.
An topological subspace inclusion is a Hurewicz cofibration precisely if is a retract of .
A subcomplex inclusion into a CW-complex is a Hurewicz cofibration
e.g. Bredon Topology and Geometry, p. 431
Every Hurewicz cofibration is an injective map and if the image is closed then it is a homeomorphism onto its image. In the category of weakly Hausdorff compactly generated spaces, is always closed (the same in the category of all Hausdorff spaces), but in the category of all topological spaces there are pathological counterexamples.
The simplest example (see the classical monograph Dieck, Kamps, Puppe, Homotopietheorie, LNM 157) is the following: let and be the one and two element sets, both with antidiscrete topology (only and are open in ), and is the inclusion . Then is a non-closed cofibration (useful exercise!).
Strøm’s model structure
make one of the standard Quillen model category structures on the category Top of all topological spaces Strøm's model category.
Interaction with pullbacks
be a commuting diagram of topological spaces such that
Then the induced morphism on pullbacks is also a closed cofibration
This is stated and proven in (Kieboom).
The product of two closed cofibrations is a closed cofibration.
Dieter Puppe, Bemerkungen über die Erweiterung von Homotopien, Arch. Math. (Basel) 18 1967 81–88; MR0206954 (34 #6770) doi
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)
The fact that morphisms of fibrant pullback diagrams along closed cofibrations induce closed cofibrations is in