Cofibrations and the Homotopy Extension Property

The following notion is one of the most classical and important in homotopy theory. 

Let us consider a special case of Definition 7.1, when i A X is an inclusion map. In this case we have a continuous map / X — y and a homotopy 

H Ax I of the restriction of / to a subspace A. The condition for the inclusion map to be a cofibration says that one should be able to extend the homotopy of A to the homotopy of the entire space X. 

Interestingly, this special case captures the whole generality of the notion of a cofibration, as our next proposition shows. 

Let us remark that so far, we have made no implicit assumptions on the topological spaces that we consider. It can be useful to know that when the space X is Hausdorff and the inclusion map i A X is a cofibration. 

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