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Elements of Point-Set Topology

  • Jewgeni H. Dshalalow
Chapter

Abstract

In Definition 4.5, Chapter 2, we called the collection of all open sets τ(d) of a metric space (X, d) the topology induced by a metric. We recall that this collection of open sets or topology is closed with respect to the formation of arbitrary unions and finite intersections. We understand that the topology of a metric space carries the main information about its structural fingerprint. For instance, equivalent metrics possess the same topology. In addition, through the topology we could establish the continuity of a function (see Theorem 4.6, Chapter 2) without need of a metric. This all leads to an idea of defining a structure more general than distance on a set, a structure that preserves convergence and continuity.

Keywords

Topological Space Open Neighborhood Open Ball Open Cover Weak Topology 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, LLC 2013

Authors and Affiliations

  • Jewgeni H. Dshalalow
    • 1
  1. 1.Mathematical SciencesFlorida Institute of TechnologyMelbourneUSA

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