Sequentially compact subsets and monotone functions: an application to fuzzy theory
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 any sequence of monotone functions from \(X\) to \(Y\) contains a pointwise convergent subsequence. The authors proceed to generalize this result in the context of \textit{admissible Hausdorff uniform structures}, as well as they establish a number of other related theorems; for several examples of LOTS, \(X\) and \(Y\), sequentially compact subsets of the space of all monotone functions from \(X\) to \(Y\) endowed with the topology of uniform convergence induced by an admissible uniformity on \(Y\) are investigated. As the authors themselves have remarked, such results apply in the space of fuzzy numbers, and further applications are expected.
scientific article; zbMATH DE number 6473617