Introduction

Ulrich Höhle
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Ulrich Höhle: Bergische Universität, Fachbereich Mathematik

A chapter in Many Valued Topology and its Applications, 2001, pp 5-9 from Springer

Abstract: Abstract Around the beginning of the 20th century General Topology arises from the effort for establishing a solid basis for Analysis and is intimately related to the success of set theory. Closed sets already appear in the work of G. Cantor; a first axiom system of neighborhoods is presented in D. Hilbert’s famous axio-matization of the plane (cf. pp. 234–235 in [39]); the notion of free ultrafilters is anticipated by F. Riesz by his definition of ideale Verdichtungsstelle (cf. [88]). One of the important contributions of F. HausdorfF consists in the freeing of the neighborhood notion from its restriction to exclusive application to higher dimensional manifolds (cf. [114]). As an illustration of this step we quote from F. Hausdorff's famous book Grundzüge der Mengenlehre (1914) the following passage: Erne Theorie der raumlichen Pimktmengen würde nun, vermöge der zahl-reichen mitspielenden Eigenschaften des gewöhnlichen Raumes, naturlich einen sehr speziellen Charakter tragen, und wenn man sich vornherein auf diesen einzigen Fall festlegen wollte, so würde man für Punktmengen einer Geraden, einer Ebene, einer Kugel usw. jedesmal eine neue Theorie zu eritwickeln haben. Die Erfahrung hat gezeigt, daß man die sen Pleonasmus vermeiden und eine allgemeinerc Thcoric aufstellen kann, die nicht nur die genannten Fälle, sondern auch noch andere Mengen (Riemannsche Flächen, Räume von endlichen und unendlich vielen Dimensionen, Kurven- und Funktionenmengen u.a.) umfafit (cf. pp. 210–211 in [37]).

Keywords: Topological Space; General Topology; Axiom System; Monoidal Category; Categorical Framework (search for similar items in EconPapers)
Date: 2001
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DOI: 10.1007/978-1-4615-1617-0_1

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