Homotopy and Isotopy Properties of Topological Spaces
The most important notion in topology is that of ahomeomorphism f: X→Yfrom a topological spaceXonto a topological spaceY. If a homeomorphism f:X→Yexists, then the topological spaces X andFare said to behomeomorphic(ortopologically equivalent), in symbols,X ≡ Y.The relation ≡ among topological spaces is obviously reflexive, symmetric, and transitive; hence it is an equivalence relation. For an arbitrary familyFof topological spaces, this equivalence relation ≡ divides /Mnto disjoint equivalence classes called thetopology typesof the familyF. Then, the main problem in topology is the topological classification problem formulated as follows.The topological classification problem:Given a familyF oftopological spaces, find an effective enumeration of the topology types of the familyFand exhibit a representative space in each of these topology types.