homotopy equivalence (original) (raw)

If there exist a homotopy equivalence between the topological spaces X and Y, we say that X and Y are_homotopy equivalent_, or thatX and Y are of the same homotopy type. We then write X≃Y.

0.0.1 Properties

1. 1. Any homeomorphism f:X→Y is obviously a homotopy equivalence withg=f-1.
3. 1. A topological space X is (by definition) contractible, if X is homotopy equivalent to a point, i.e., X≃{x0}.

References

• 1 A. Hatcher, Algebraic Topology, Cambridge University Press, 2002. Also availablehttp://www.math.cornell.edu/ hatcher/AT/ATpage.htmlonline.