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Homotopy and Isotopy Properties of Topological Spaces
Published online by Cambridge University Press: 20 November 2018
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The most important notion in topology is that of a homeomorphism f: X → Y from a topological space X onto a topological space Y. If a homeomorphism f: X → Y exists, then the topological spaces X and F are said to be homeomorphic (or topologically 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 family F of topological spaces, this equivalence relation ≡ divides /Mnto disjoint equivalence classes called the topology types of the family F. Then, the main problem in topology is the topological classification problem formulated as follows.
The topological classification problem: Given a family F of topological spaces, find an effective enumeration of the topology types of the family F and exhibit a representative space in each of these topology types.
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