homotopy equivalence

Definition

Let , be topological spaces. We call these spaces homotopy equivalent given there exists two maps and with homotopies.

Notes

Homotopy equivalence is weaker than homeomorphism. Homeomorphism implies homotopy, since we can take the homotopies to be constant of the inverse maps.

and are homotopy equivalent, but are also homotopy equivalent, so homotopy equivalence doesn’t preserve compactness.

Any star convex subset is homotopy equivalent to a point. We can take the homotopies to be the contraction down to a point and the “inverse” contraction that is an inclusion into .