Elements of Point-Set Topology

  • Jewgeni H. Dshalalow


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.


Topological Space Open Neighborhood Open Ball Open Cover Weak Topology 
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© Springer Science+Business Media, LLC 2013

Authors and Affiliations

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

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