# Cofibration

(Redirected from Closed inclusion)

In mathematics, in particular homotopy theory, a continuous mapping

$i\colon A \to X$,

where A and X are topological spaces, is a cofibration if it satisfies the homotopy extension property with respect to all spaces Y. The name is because the dual condition, the homotopy lifting property, defines fibrations. For a more general notion of cofibration see the article about model categories.

## Basic theorems

• For Hausdorff spaces a cofibration is a closed inclusion (injective with closed image); for suitable spaces, a converse holds
• Every map can be replaced by a cofibration via the mapping cylinder construction
• There is a cofibration (A, X), if and only if there is a retraction from
$X \times I$
to
$(A \times I) \cup (X \times \{0\})$,

since this is the pushout and thus induces maps to every space sensible in the diagram.

## Examples

• Cofibrations are preserved under push-outs and composition, as one sees from the definition via diagram-chasing.
• A frequently used fact is that a cellular inclusion is a cofibration (so, for instance, if $(X, A)$ is a CW pair, then $A \to X$ is a cofibration). This follows from the previous fact since $S^{n-1} \to D^n$ is a cofibration for every $n$.