The Homotopy Extension Property (HEP) is a fundamental concept in algebraic topology. It states that given a continuous map between two topological spaces, if the map can be extended to a homotopy between the two spaces, then it can be extended to a homotopy between any larger spaces that contain the original two spaces as subspaces.
The Homotopy Extension Property allows us to study topological spaces by looking at their subspaces, rather than the spaces themselves. This greatly simplifies the study of complicated spaces and helps to uncover important algebraic and geometric properties.
The Homotopy Extension Property has numerous applications in mathematics, particularly in algebraic topology and differential geometry. It is used to prove important theorems, such as the Hurewicz theorem and the Whitehead theorem, and to study fundamental groups, homotopy groups, and higher homotopy groups.
Some common examples of spaces that satisfy the Homotopy Extension Property include contractible spaces, simply-connected spaces, and CW complexes. These spaces have simple topological structures that make it easy to extend homotopies between subspaces.
Yes, there are known examples of spaces that do not satisfy the Homotopy Extension Property, such as the Hawaiian earring space. These spaces often have more complicated topological structures that make it difficult to extend homotopies between subspaces.