Sequentially compact subsets and monotone functions: an application to fuzzy theory
type: Item
instance of: scholarly article
title: Sequentially compact subsets and monotone functions: an application to fuzzy theory
zbMATH Open document ID: 1325.54014
author: Juan J. Font
author: Manuel Sanchis
published in: Topology and its Applications
publication date: 2015-08-19T00:00:00Z
review text: In what follows, the \textit{weight} of a topological space \(X\) (denoted \(w(X)\)) is the smallest cardinality of a base of \(X\). A family \(\mathcal{S}\) of infinite subsets of \(\mathbb{N}\) is \textit{splitting} if for every infinite subset \(A \subseteq \mathbb{N}\) there exists a set \(S \in \mathcal{S}\) such that \(A \cap S\) and \(A \setminus S\) are both infinite. The \textit{splitting number} \(\mathfrak{s}\) is defined as the smallest cardinality of a splitting family. The small cardinal \(\mathfrak{s}\) is easily seen to be uncountable, and both equalities \(\aleph_1 = \mathfrak{s}\) and \(\mathfrak{s} = \mathfrak{c}\) are consistent with \textbf{ZFC} and independent of the Continuum Hypothesis. The paper under review investigates linearly ordered topological spaces -- LOTS, for short -- which are the topological spaces induced by linear orders (i.e., \((X,<,\tau)\) is a LOTS if \(\tau\) is the topology generated over \(X\) by the collection of all open intervals of \((X,<)\), where \(<\) is a linear order on \(X\)). In this class of spaces, convergence of sequences can be defined in a canonical way: a nondecreasing (respectively, nonincreasing) sequence \((x_n)_{n \in \mathbb{N}}\) converges to \(x_0\) if, and only if, \(x_0\) is the supremum (respectively, the infimum) of the set of all elements of the sequence. In general, a sequence \((x_n)_{n \in \mathbb{N}}\) converges to \(x_0\) if, and only if, every monotone subsequence converges to \(x_0\). Since every sequence in a linearly ordered set contains a monotone subsequence, we have that a LOTS \((X,<,\tau)\) is \textit{sequentially compact} (i.e., every sequence in \(X\) has a convergent subsequence) if, and only if, every monotone sequence converges. It was previously shown by \textit{S. Fuchino} and \textit{S. Plewik} [Proc. Am. Math. Soc. 127, No. 2, 491--497 (1999; Zbl 0908.26004)] that if \((X,<_X)\) and \((Y,<_Y)\) are LOTS and \(Y\) is sequentially compact with \(w(Y) < \mathfrak{s}\) then…
reviewed by: Samuel Gomes da Silva
Mathematics Subject Classification ID: 54F05
Mathematics Subject Classification ID: 03E72
zbMATH DE Number: 6473617
zbMATH Keywords: linearly ordered topological spaces
zbMATH Keywords: monotone functions
zbMATH Keywords: sequential compactness
zbMATH Keywords: fuzzy number
MaRDI profile type: Publication
full work available at URL: j.topol.2015.05.074
OpenAlex ID: W1029750992
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DOI: 10.1016/J.TOPOL.2015.05.074
description: scientific article; zbMATH DE number 6473617