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Topological Structures

In: The Elements of Operator Theory

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  • Carlos S. Kubrusly

    (Catholic University of Rio de Janeiro, Electrical Engineering Department)

Abstract

The basic concept behind the subject of point-set topology is the notion of “closeness” between two points in a set X. In order to get a numerical gauge of how close together two points in X may be, we shall provide an extra structure to X, viz., a topological structure, that again goes beyond its purely settheoretic structure. For most of our purposes the notion of closeness associated with a metric will be sufficient, and this leads to the concept of “metric space”: a set upon which a “metric” is defined. The metric-space structure that a set acquires when a metric is defined on it is a special kind of topological structure. Metric spaces comprise the kernel of this chapter, but general topological spaces are also introduced.

Suggested Citation

  • Carlos S. Kubrusly, 2011. "Topological Structures," Springer Books, in: The Elements of Operator Theory, chapter 3, pages 87-198, Springer.
    • Handle: RePEc:spr:sprchp:978-0-8176-4998-2_3
    DOI: 10.1007/978-0-8176-4998-2_3
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