Separation properties at \(p\) for the topological category of Cauchy spaces
type: Item
instance of: scholarly article
title: Separation properties at \(p\) for the topological category of Cauchy spaces
zbMATH Open document ID: 1274.54048
author: Muammer Kula
published in: Acta Mathematica Hungarica
publication date: 2012-09-17T00:00:00Z
review text: Cauchy spaces, designed to provide a general setting for studying classical Cauchy convergence, are pairs \(\langle X,\mathcal C\rangle\), where \(\mathcal C \) is a family of proper filters on a set \(X\) -- the Cauchy filters on \(X\) -- satisfying: (1) the filter of supersets of a point is Cauchy; (2) a filter containing a Cauchy filter is also Cauchy; and (3) if \(F\) and \(G\) are Cauchy and each member of \(F\) intersects each member of \(G\), then \(F\cap G\) is Cauchy. A function from a Cauchy space \(\langle X,\mathcal C\rangle\) to a Cauchy space \(\langle Y,\mathcal D\rangle\) is Cauchy continuous if the image of each filter in \(\mathcal C\) generates a filter in \(\mathcal D\). These ingredients constitute the topological category \textbf{CHY}. In a series of papers \textit{M. Baran} [Math.\ Balk., New Ser. 10, No. ~1, 39--48 (1996; Zbl 1036.54502)] defined separation properties at a point, and generalized this to point-free definitions via the generic element method of topos theory for an arbitrary topological category over sets. In the present paper, it is the author's stated goal to: (1) give a characterization for each of the generalized versions of T\({}_i\) \((i = 0, 1, 2)\) and Pre T\({}_2\) at a point in \textbf{CHY}; (2) to examine how these generalizations are related; and (3) to show that each of the generalized (point-free) separation properties implies its corresponding version at a point in \textbf{CHY}.
reviewed by: Paul Bankston
Mathematics Subject Classification ID: 54B30
Mathematics Subject Classification ID: 54D10
Mathematics Subject Classification ID: 54A05
Mathematics Subject Classification ID: 54A20
Mathematics Subject Classification ID: 18B99
zbMATH DE Number: 6083179
zbMATH Keywords: topological category
zbMATH Keywords: Cauchy space
zbMATH Keywords: Cauchy map
zbMATH Keywords: separation
zbMATH Keywords: generalized separation properties
zbMATH Keywords: separation properties at a point
zbMATH Keywords: Cauchy spaces
MaRDI profile type: Publication
full work available at URL: s10474-012-0238-z
OpenAlex ID: W2112655894
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DOI: 10.1007/S10474-012-0238-Z
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description: scientific article; zbMATH DE number 6083179