Variations on a theorem of Arhangel'skii and Pytkeev
Arkhangel'skij and Pytkeev proved (for details, see \textit{A. V. Arkhangel'skij} [Topological Function Spaces. Mathematics and its Applications, Soviet Series 78. Dordrecht etc.: Kluwer Academic Publishers (1992; Zbl 0758.46026)]) that the tightness of the space \(C_p(X)\) equals the supremum of the Lindelöf numbers of finite powers of \(X\). If \(\text{Exp}(X)\) denotes the set of all closed subsets of a space \(X\), then the lower Vietoris topology on \(\text{Exp}(X)\) is the one having as a base the family of all subsets of the form \(\{F \in\text{Exp}(X) : F\cap U \neq\emptyset\;\text{for every}\;U\in \mathbb V\}\) where \(\mathbb V\) is a finite set of open subsets of \(X\). The lower Vietoris topology on \(C(X)\), the set of all real functions on an infinite Tychonoff space \(X\) is the one obtained by inheritance from \(\text{Exp}(X \times \mathbb R)\). The point-open topology (or pointwise convergence topology) on \(C(X)\) is the one generated by the family of all sets of the form \(\{f \in C(X): f[M]\subseteq U\}\) where \(M\) ranges over finite subsets of \(X\) and \(U\) over open sets of \(\mathbb R\). For two topologies \(\tau_1\), \(\tau_2\) on \(X\), with \(\tau_1\supseteq\tau_2\), the tightness of \(\tau_1\) relative to \(\tau_2\) is defined to be the least infinite cardinal \(\kappa\) such that whenever \(x\in X\), \(A\subseteq X\), if \(x\in cl_{\tau1}(A)\), then there is a \(B \subseteq A\) of cardinality \(\leq \kappa\) such that \(x\in cl_{\tau2}(B)\). The weak Lindelöf number of a space, \(wl(X)\) is defined as the least infinite cardinal \(\kappa\) such that for each open cover \(V\) of \(X\), there is a \(\mathbb U\subset \mathbb V\) of cardinality \(\leq \kappa\) such that \(cl(\cup\mathbb U) = X\). (A family like this is called a weak cover of \(X\)). Define the following selection principle \(S_{fin}(\mathbb A ,\mathbb B)(S_1(\mathbb A,\mathbb B)\) respectively): for each sequence \(\langle A_n:n\in \mathbb N\rangle\) of elements of \(\mathbb A\), there is a sequence \(\langle B_n:n\in \mathbb N\rangle\) where \(B_n\) is a finite set (singleton respectively) with \(B_n\subseteq A_n\) and \(\cup\{B_n:n \in \mathbb N\}\in \mathbb B\). In this paper, the author calculates a bitopological version of tightness using the weak Lindelöf numbers of finite powers of \(X\). Also he characterizes bitopological versions of countable fan and strong fan tightness of the point-open topology with respect to the lower Vietoris topology on \(C(X)\) in terms of suitable covering properties of the powers of \(X\) formulated using the language of \(S_1\) and \(S_{fin}\) selection principles.
scientific article; zbMATH DE number 5990614
scientific article