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.
Let $ X $ be a topological space, and let $ A \subset X $. We say that the pair $ ( X , A ) $ has the homotopy extension property if, given a homotopy $ f _ t : A \to Y $ and a map $ F _ 0 : X \to Y $ such that $ { F _ 0 } | _ A = f _ 0 $, there exists an extension of $ F _ 0 $ to a homotopy $ F _ t : X \to Y $ such that $ { F _ t } | _ A = f _ t $.
That is, the pair $ ( X , A ) $ has the homotopy extension property if any map $ G : \big( ( X \times \lbrace 0 \rbrace ) \cup ( A \times I ) \big) \to Y $ can be extended to a map $ G ′ : X \times I \to Y $ (i.e. $ G $ and $ G ′ $ agree on their common domain).
If the pair has this property only for a certain codomain $ Y $, we say that $ ( X , A ) $ has the homotopy extension property with respect to $ Y $.
A pair $ ( X , A ) $ has the homotopy extension property if and only if $ \big( ( X \times \lbrace 0 \rbrace ) \cup ( A \times I ) \big) $ is a retract of $ X \times I $.
If $ X $ is a cell complex and $ A $ is a subcomplex of $ X $, then the pair $ ( X , A ) $ has the homotopy extension property.
If $ ( X , A ) $ has the homotopy extension property, then the simple inclusion map $ i : A \to X $ is a cofibration. In fact, if you consider any cofibration $ i : Y \to Z $, then we have that $ Y $ is homeomorphic to its image under $ i $. This implies that any cofibration can be treated as an inclusion map, and therefore it can be treated as having the homotopy extension property.
The homotopy extension property of cofibrations is dual to the homotopy lifting property that is used to define fibrations.
Source: Wikipedia
