Homotopy extension property

In mathematics, in the area of algebraic topology, the homotopy extension property indicates which homotopies defined on a subspace can be extended to a homotopy defined on a larger space. The homotopy extension property of cofibrations is dual to the homotopy lifting property that is used to define fibrations.

Let ${\displaystyle X\,\!}$ be a topological space, and let ${\displaystyle A\subset X}$. We say that the pair ${\displaystyle (X,A)\,\!}$ has the homotopy extension property if, given a homotopy ${\displaystyle f_{t}\colon A\rightarrow Y}$ and a map ${\displaystyle F_{0}\colon X\rightarrow Y}$ such that ${\displaystyle F_{0}|_{A}=f_{0}}$, there exists an extension of ${\displaystyle F_{0}}$ to a homotopy ${\displaystyle F_{t}\colon X\rightarrow Y}$ such that ${\displaystyle F_{t}|_{A}=f_{t}}$.

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