The Mathematical Structure of Classical and Relativistic Physics by Enzo Tonti

Author:Enzo Tonti
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

7.2 Topology and Algebraic Topology

Topology can be defined as the science which studies the properties of geometric figures which are preserved under continuous deformations without tearing or overlaps. Since a deformation can be viewed as a transformation, the transformations considered by topology are continuous, invertible and with continuous inverses; for such transformations Poincaré introduced the term homeomorphism. Thus, briefly put, topology studies the properties of geometric figures which are invariant under homeomorphisms.

The subject of topology deals with those properties of geometric figures which are unchanged by topological mappings, that is, by mappings which are bijective (i.e. one-to-one correspondences) and bicontinuous (i.e. continuous, with continuous inverses). Those properties which remain unchanged under topological mappings are called the topological properties of the figures. Two figures which can be mapped topologically onto each other are said to be homeomorphic.1

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