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 $i \colon A \longrightarrow X$ is a Hurewicz cofibration if it satisfies the homotopy extension property in that:
for any topological space $Y$,
all continuous functions $f \colon A\to Y$, $\tilde{f}:X\to Y$ such that $\tilde{f}\circ i=f$
and any left homotopy $F \colon A\times I\to Y$ such that $F(-,0)=f$
there is a homotopy $\tilde{F}:X\times I\to Y$ such that
$\tilde{F}\circ(i\times id_I)=F$
and $\tilde{F}(-,0)=\tilde{f}$.
A Hurewicz cofibration $i:A\to X$ (def. 1) is called a closed cofibration if the image $i(A)$ is a closed subspace in $X$.
If $A\subset X$ is closed and the inclusion is a cofibration, then the pair $(X,A)$ is called an NDR-pair.
There is also a version of the definition for pointed spaces.
An topological subspace inclusion $A \hookrightarrow X$ is a Hurewicz cofibration precisely if $A \times I \cup X \times \{0\}$ is a retract of $X\times I$.
A subcomplex inclusion into a CW-complex is a Hurewicz cofibration
e.g. Bredon Topology and Geometry, p. 431
Every Hurewicz cofibration $i$ is an injective map and if the image $i(A)$ is closed then it is a homeomorphism onto its image. In the category of weakly Hausdorff compactly generated spaces, $i(A)$ 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 $A =\{a\}$ and $X=\{a,b\}$ be the one and two element sets, both with antidiscrete topology (only $X$ and $\emptyset$ are open in $X$), and $i:A\hookrightarrow X$ is the inclusion $a\mapsto a$. Then $i$ is a non-closed cofibration (useful exercise!).
The collections
make one of the standard Quillen model category structures on the category Top of all topological spaces Strøm's model category.
Let
be a commuting diagram of topological spaces such that
the horizontal morphisms are closed cofibrations;
the morphisms $p_0$ and $p$ are Hurewicz fibrations.
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