Homotopy Type of Topological Spaces

Theorem 6.4 motivates the introduction of the corresponding equivalence relation on the topological spaces themselves. 

Definition 6.5. Two topological spaces are called homotopy equivalent if there exist continuous maps 95 X — X and ip Y X such that ipoip idy and -0 o 95 idx. In that case, we write X c lY. 

Indeed, Theorem 6.4 implies the following fundamental fact, strengthening the homeomorphism invariance that was previously mentioned in Remark 3.16. 

Theorem 6.6. Homology groups are homotopy invariants in other words, if X and Y are homotopy equivalent CW complexes, then iJ (X) = Hn(Y), for any n 0. 

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