K-8 - Homotopy, II

Abstract

Considering a pair (X, A) and a space Y, when two homotopic maps admit continuous extensions over X, a question may arise as to whether the extensions can be homotopic and under what condition. This question suggests the homotopy extension problem. When the question is affirmatively solved for every space Y and map H, it can be said that the pair (X, A) has the homotopy extension property or the inclusion map : A→X is a cofibration. For example, a pair (K, L) of a CW-complex K and its subcomplex L and a pair (X, A) of an absolute neighborhood retract (ANR) X and its closed subset A that is an ANR has the homotopy extension property. When the question is affirmatively solved for every map H, it can be said that Y has the homotopy extension property with respect to (X, A). Every ANR has the homotopy extension property with respect to all pairs (X, A) of a normal and countably paracompact space X and its closed subset A. This homotopy extension property introduces an important notation in general topology called Dowker spaces.