Continuous Transformations and Equivalent Figures

Topology is concerned with those properties of geometric figures that are invariant under continuous transformations. A continuous transformation, also called a topological transformation or homeomorphism, is a one-to-one correspondence between the points of one figure and the points of another figure such that points that are arbitrarily close on one figure are transformed into points that are also arbitrarily close on the other figure. Figures that are related in this way are said to be topologically equivalent. If a figure is transformed into an equivalent figure by bending, stretching, etc., the change is a special type of topological transformation called a continuous deformation. Two figures (e.g, certain types of knots) may be topologically equivalent, however, without being changeable into one another by a continuous deformation.

It is intuitively evident that all simple closed curves in the plane and all polygons are topologically equivalent to a circle; similarly, all closed cylinders, cones, convex polyhedra, and other simple closed surfaces are equivalent to a sphere. On the other hand, a closed surface such as a torus (doughnut) is not equivalent to a sphere, since no amount of bending or stretching will make it into a sphere, nor is a surface with a boundary equivalent to a sphere, e.g., a cylinder with an open top, which may be stretched into a disk (a circle plus its interior).

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The Columbia Electronic Encyclopedia, 6th ed. Copyright © 2012, Columbia University Press. All rights reserved.

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