Cofibration

Builds on Homotopy Homotopy extension property Continuous map

Definition

Let $A$ and $X$ be topological spaces and let $I = [0, 1]$ . Let $i \colon A \to X$ be a continuous map. The map $i$ is a cofibration if the following holds for every topological space $Y$ . For every continuous map $f \colon X \to Y$ and every homotopy $h \colon A \times I \to Y$ with $h(a, 0) = f(i(a))$ for all $a \in A$ , there is a homotopy $H \colon X \times I \to Y$ with:

$H(x, 0) = f(x)$ for all $x \in X$ ;
$H(i(a), t) = h(a, t)$ for all $a \in A$ and $t \in I$ .

Remarks

For the inclusion of a subspace $A \subseteq X$ , this says that the pair $(X, A)$ has the homotopy extension property. Every cofibration is a homeomorphism onto its image, and if $X$ is Hausdorff the image is closed in $X$ . Example: the inclusion $S^{n-1} \to D^n$ is a cofibration, and so is the inclusion of a subcomplex of a CW complex. Non-example: the inclusion $(0, 1) \to [0, 1]$ is not a cofibration, since $[0, 1]$ is Hausdorff and $(0, 1)$ is not closed in it. Cofibrations are stable under pushout and under composition.

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Continuous map concept
Homotopy concept
Homotopy extension property concept
Cofibration definition

Tag	`EFJ77PFF88`
URI	`m2://EFJ77PFF88`
Kind	definition
Writing	W0 stub
MSC	55P05
Defines	Cofibration
Uses	Homotopy, Homotopy extension property, Continuous map
Changed	2 November 2026

Cofibration

Definition

Remarks

Reading path

Further reading

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