We apply the theory of infinite two-person games to two well-known problems in topology: Suslin’s Problem and Arhangel’skii’s problem on the weak Lindelöf number of the $G_{\delta }$ topology on a compact space. More specifically, we prove results of which the following two are special cases: 1) every linearly ordered topological space satisfying the game-theoretic version of the countable chain condition is separable, and 2) in every compact space satisfying the game-theoretic version of the weak Lindelöf...

This work presents some cardinal inequalities in which appears the closed pseudo-character, ${\psi }_c$, of a space. Using one of them — ${\psi }_c\left(X\right)\le 2^{d\left(X\right)}$ for $T_2$ spaces — we improve, from $T_3$ to $T_2$ spaces, the well-known result that initially $\kappa$-compact $T_3$ spaces are $\lambda$-bounded for all cardinals $\lambda$ such that $2^{\lambda }\le \kappa$. And then, using an idea of A. Dow, we prove that initially $\kappa$-compact $T_2$ spaces are in fact compact for $\kappa =2^{F\left(X\right)}$, $2^{s\left(X\right)}$, $2^{t\left(X\right)}$, $2^{\chi \left(X\right)}$, $2^{{\psi }_c\left(X\right)}$ or $\kappa =max\{{\tau }^{+},{\tau }^{<\tau }\}$, where $\tau >t(p,X)$ for all $p\in X$.

A family of subsets of a set is called a $\sigma$-topology if it is closed under arbitrary countable unions and arbitrary finite intersections. A $\sigma$-topology is perfect if any its member (open set) is a countable union of complements of open sets. In this paper perfect $\sigma$-topologies are characterized in terms of inserting lower and upper measurable functions. This improves upon and extends a similar result concerning perfect topologies. Combining this characterization with a $\sigma$-topological version of Katětov-Tong...

We call a topological space $\kappa$-compact if every subset of size $\kappa$ has a complete accumulation point in it. Let $\Phi (\mu ,\kappa ,\lambda )$ denote the following statement: $\mu <\kappa <\lambda =cf\left(\lambda \right)$ and there is $\{S_{\xi }:\xi <\lambda \}\subset {\left[\kappa \right]}^{\mu }$ such that $\left|\right\{\xi :|S_{\xi }\cap A|=\mu \left\}\right|<\lambda$ whenever $A\in {\left[\kappa \right]}^{<\kappa }$. We show that if $\Phi (\mu ,\kappa ,\lambda )$ holds and the space $X$ is both $\mu$-compact and $\lambda$-compact then $X$ is $\kappa$-compact as well. Moreover, from PCF theory we deduce $\Phi (cf\left(\kappa \right),\kappa ,{\kappa }^{+})$ for every singular cardinal $\kappa$. As a corollary we get that a linearly Lindelöf and ${\aleph }_{\omega }$-compact space is uncountably compact, that is $\kappa$-compact for all uncountable cardinals $\kappa$.

Given two topologies, $T_1$ and $T_2$, on the same set X, the intersection topologywith respect to $T_1$ and $T_2$ is the topology with basis $U_1\cap U_2:U_1\in T_1,U_2\in T_2$. Equivalently, T is the join of $T_1$ and $T_2$ in the lattice of topologies on the set X. Following the work of Reed concerning intersection topologies with respect to the real line and the countable ordinals, Kunen made an extensive investigation of normality, perfectness and ${\omega }_1$-compactness in this class of topologies. We demonstrate that the majority of his results generalise...

We start by analyzing the role of imprecision in information retrieval in the Web, some theoretical contributions for managing this problem and its presence in search engines, with special emphasis on the use of thesaurus in order to increase the relevance of the documents retrieved. We then present FDSA, a Spanish electronic dictionary of synonyms that compute degrees of synonymy, and an eflicient implementation of it by using deterministic acyclic finite-state automata. We conclude by conjecturing...

The main purpose of this paper is to introduce the concept of intuitionistic $\mathrm{I}$-fuzzy quasi-coincident neighborhood systems of intuitiostic fuzzy points. The relation between the category of intuitionistic $I$-fuzzy topological spaces and the category of intuitionistic $I$-fuzzy quasi-coincident neighborhood spaces are studied. By using fuzzifying topology, the notion of generated intuitionistic $I$-fuzzy topology is proposed, and the connections among generated intuitionistic $I$-fuzzy topological spaces,...

We establish results on invariant approximation for fuzzy nonexpansive mappings defined on fuzzy metric spaces. As an application a result on the best approximation as a fixed point in a fuzzy normed space is obtained. We also define the strictly convex fuzzy normed space and obtain a necessary condition for the set of all $t$-best approximations to contain a fixed point of arbitrary mappings. A result regarding the existence of an invariant point for a pair of commuting mappings on a fuzzy metric...