Abstract
Topology studies those characteristics of figures which are preserved under a certain class of continuous transformations. Imagine two figures, a square and a circular disk, made of rubber. Deformations can convert the square into the disk, but without tearing the figure it is impossible to convert the disk by any deformation into an annulus. In topology, this intuitively obvious distinction is formalized. Two figures which can be transformed into one other by continuous deformations without cutting and pasting are called homeomorphic. For example, the totality of sides of any polygon is homeomorphic to a circle, but a circle is not homeomorphic to a straightline segment; a sphere is homeomorphic to a closed cylinder but not to a torus, and so on.
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References
For expository accounts of their work, see: William P. Thurston, “Three dimensional manifolds, Kleinian groups, and hyperbolic geometry,” Bull. Amer. Math. Soc. (NS), 6 (1982), 357–381
and Michael H. Freedman, “There is no room to spare in four-dimensional space,” Amer. Math. Soc. Notices 31 (1984), 3–6.
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© 1999 Springer Science+Business Media New York
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Monastyrsky, M. (1999). Topological Structures. In: Riemann, Topology, and Physics. Modern Birkhäuser Classics. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-4779-7_9
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DOI: https://doi.org/10.1007/978-0-8176-4779-7_9
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