Why is the indiscreet topology important

Set theoretical topology

Algebraic Topology pp 9-65 | Cite as

Summary

In metric spaces, the spatial relationship of points is described by their distance from one another. With the help of the metric, the concept of environment is introduced and open and closed sets are defined. These open sets have the following characteristic properties: The union of any number of open sets is open, and the intersection of finitely many open sets is also open. These two properties turned out to be fundamental for an axiomatic description of spatial relationships. Becomes from alien subsets of a set X denotes a system of subsets which is closed with respect to the formation of finite averages and arbitrary unions, and in addition X itself and contains the empty set, this can be broken down as a system of open sets X consider. Such a system is called topology X, and some of the definitions and theorems in connection with open sets, which are known from the introductory courses on calculus, can be applied to the set Xon which a topology is marked. So can each point of X assign a system of environments, and with the help of the concept of environment, the concepts of open core, closed shell and edge of a subset can be transferred directly to the abstract set with a topology. The same applies to the concept of continuous mapping.

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Copyright information

© Birkhäuser Verlag Basel 1989

Authors and Affiliations

  1. 1. Department of Mathematics, University of Dortmund, Dortmund 50, Germany