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Elements of Point-Set Topology

In: Foundations of Abstract Analysis

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  • Jewgeni H. Dshalalow

    (Florida Institute of Technology, Mathematical Sciences)

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.

Suggested Citation

  • Jewgeni H. Dshalalow, 2013. "Elements of Point-Set Topology," Springer Books, in: Foundations of Abstract Analysis, chapter 0, pages 171-236, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4614-5962-0_3
    DOI: 10.1007/978-1-4614-5962-0_3
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