Topological Spaces

Erdoğan S. Şuhubi
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Erdoğan S. Şuhubi: Yeditepe University, Department of Mathematics

Chapter Chapter IV in Functional Analysis, 2003, pp 221-260 from Springer

Abstract: Abstract While we were discussing, in the preceding chapter, some fundamental concepts associated with real analysis we observed that concepts such as convergence, limit, continuity of functions, compactness or denseness can be properly defined and effectively exploited by employing certain subsets of the set of real numbers with appropriate properties which we have called open sets that are essentially unions of open intervals. In this chapter, inspired by that approach, we try to extend some concepts developed in real analysis to an arbitrary set X through an appropriately chosen class of its subsets enabling us to introduce a kind of notion of “nearness” between objects in this set. This class then defines a topology on X. Sets equipped with topologies will be called topological spaces. We shall briefly discuss here their structures and various properties sufficiently to meet our demands in the future chapters.

Keywords: Topological Space; Open Neighbourhood; Open Cover; Accumulation Point; Topological Vector Space (search for similar items in EconPapers)
Date: 2003
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-94-017-0141-9_4

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DOI: 10.1007/978-94-017-0141-9_4

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