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Homotopy and Isotopy Properties of Topological Spaces

Published online by Cambridge University Press:  20 November 2018

Sze-Tsen Hu
Sze-Tsen Hu*
Affiliation:
University of California at Los Angeles
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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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 13 , 1961 , pp. 167 - 176
Copyright
Copyright © Canadian Mathematical Society 1961

References

1. Hu, S. T., Isotopy invariants of topological spaces, Technical Note, AFORS TN59-236, AD212006. Also to appear in the Proceedings of the Royal Society, England.Google Scholar
2. Hurewicz, W. and Wallman, H., Dimension Theory (Princeton Mathematical Series, No. 4) Princeton University Press, 1941]).Google Scholar
3. Postnikov, M. M., Investigations in homotopy theory of continuous mappings I, II, Trudy Mat. Inst. Steklov, No. 46, Izdat. Akad. Nauk SSSR (Moskow, 1955). Amer. Math. Soc. Translations, Series 2, 7 (1957).Google Scholar
4. Shapiro, A., Obstructions to the imbedding of a complex in a euclidean space I, Ann. Math., 66 (1957), 256–269.Google Scholar
5. Wu, W. T., On the realization of complexes in Euclidean spaces I-III, Acta Math. Sinica, 5 (1955), 505-552; 7 (1957), 79-101; 8 (1958), 79–94.Google Scholar
6. Wu, W. T., On the imbedding of polyhedra in Euclidean spaces, Bull. Polon. Sci. CI. III, 4 (1956), 573–577.Google Scholar