Elements of Point-Set Topology

Jewgeni H. Dshalalow
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Jewgeni H. Dshalalow: Florida Institute of Technology, Mathematical Sciences

Chapter Chapter 3 in Foundations of Abstract Analysis, 2013, pp 171-236 from Springer

Abstract: 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 (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/978-1-4614-5962-0_3

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