A property of an inclusion i:A→X (or of the pair (X,A)) that any homotopy H_t defined on A can be extended to a homotopy ˜H_t on X starting from any map f:X→Y restricting to the given homotopy on A; equivalently a recognition criterion for cofibrations in many settings.

Principle

Control extensions by local deformation data: if A sits in X as a cofibration (or an NDR-pair), homotopies on A admit extensions to X because one can thicken A and push the homotopy outwards along a cylinder or deformation retraction.

Demonstration

In CW complexes the inclusion of a subcomplex A↪X has the HEP: a homotopy on A extends cell-by-cell to X using the cellular attaching maps and the fact cells are attached by cofibrations, allowing induction over skeleta.

Misapplication

Assuming arbitrary subspace inclusions have HEP leads to false extensions (for instance pathological embeddings in bad point-set topologies); conflating HEP with homotopy lifting property (HLP) for fibrations is another common mistake.

Consequence

When a pair has HEP one can construct extensions of maps and homotopies, identify cofibrations, form homotopy pushouts, and apply obstruction-theoretic arguments for extending structures stepwise.

Reversal

The dual or opposite situation is the homotopy lifting property (HLP) for fibrations, where homotopies lift along a map rather than extend from a subspace; or simply inclusions that obstruct extension so that local homotopies cannot be globally extended.

Boundary

Pertains to inclusions of subspaces and topological pairs; it presumes reasonable categories (CW complexes, compactly generated spaces, or spaces where cofibrations are well behaved) and excludes arbitrary maps that are not inclusions or settings without cylinders.

Semantic Tension

Tension exists between HEP and HLP (extension versus lifting), and between the intuitive idea that subspaces extend homotopies and counterexamples in pathological point-set contexts; one must distinguish cofibration hypotheses from weaker inclusion properties.

Synthesis

HEP identifies inclusions that allow homotopies defined on a subspace to be extended to the whole space; in practice it characterizes cofibrations and underpins inductive and obstruction-theoretic constructions in homotopy theory.