Calibers, free sequences and density (Q1602958)

This paper provides some interesting results concerning the influence of several notions of caliber on the density of a space that is a union of a ``small'' number of compact sets which have no ``long'' free sequences. A cardinal \(\kappa\) is said to be a caliber of a space \(X\) (\(\kappa\in Cal(X)\)) provided that among any family of \(\kappa\) many open sets there is a subfamily of cardinality \(\kappa\) that has non-empty intersection. Obviously if the cofinality of \(\kappa\) is larger than the density of \(X\) (\(cf(\kappa) > d(X)\)), then \(\kappa\in Cal(X)\), but the converse in not true in general. This paper is concerned with conditions which will imply the converse, or at least provide an upper bound for \(d(X)\). A sequence \(\{x_\alpha:\alpha<\kappa\}\) is called a free sequence (of length \(\kappa\)) provided that for all \(\gamma <\kappa\), \(cl_X(\{x_\alpha:\alpha<\gamma\})\cap cl_X( \{x_\alpha:\gamma\leq \alpha<\kappa\})=\emptyset\) (the concept is due to A. Arhangel\('\)skii). The authors prove: if \(X\) is a \(T_3\)-space, \(\kappa\in Cal(X)\), and \(X\) is the union of at most \(\kappa\) many compact subsets each with no free sequences of length \(\kappa\), then \(d(X) <\kappa\). Since countable tightness implies there are no uncountable free sequences, a new corollary to the previous result states: If \(\omega_1\) is a caliber of \(X\) and \(X\) is the union of \(\omega_1\) compact sets each of countable tightness, then \(X\) is separable. The authors say that \((\lambda,\kappa)\) is a (pair) caliber of \(X\) (\((\lambda,\kappa)\in Cal_2(X)\)) if \(\lambda\leq \kappa\) and among any family of \(\kappa\) many open sets there is a subfamily of cardinality \(\lambda\) with non-empty intersection. The authors prove: If \(X\) is a \(T_3\)-space and \((\mu,\kappa)\) is a caliber of \(X\), and \(X\) is the union of at most \(\kappa\) many compact sets each having no free sequences of length \(\mu\), then \(d(X) <\kappa\). The authors say that \((\mu,\lambda,\kappa)\) is a (triple) caliber of \(X\) (\((\mu,\lambda,\kappa)\in Cal_3(X)\)) if \(\mu\leq \lambda\leq \kappa\) and among any family of \(\kappa\) many open sets there is a subfamily of cardinality \(\lambda\) such that each of its subfamilies of cardinality less than \(\mu\) has non-empty intersection. The authors prove: If \(X\) is a \(T_3\)-space, \((\lambda,\lambda,\kappa)\) is a caliber of \(X\), and \(X\) has no free sequences of length \(\lambda\), then \(d(X) \leq\mu <\lambda\) for some cardinal \(\mu <\kappa\). The results strengthen results of \textit{B. Shapirovksii} [see [3.25] in the book by the first author ``Cardinal functions in topology -- ten years later'', Math. Center Tract, Vol. 123, North-Holland, Amsterdam (1980; Zbl 0479.54001)] and generalize results of \textit{A. Arhangel'skii} [Topology Appl. 104, No. 1-3, 13-16 (2000; Zbl 0944.54012)]. Reviewer's remark: Some historical notes on caliber and related topics, and similar notions of double and triple caliber, can be found in the book by \textit{W. W. Comfort} and \textit{S. Negrepontis} [Chain Conditions in Topology, Cambridge Univ. Press (1982; Zbl 0488.54002)]. The concept of caliber was introduced by N. A. Shanin in 1942. The concept of pair caliber and a version of triple caliber that corresponds to \((\mu^+,\lambda,\kappa)\) caliber above were introduced by W. W. Comfort and S. Negrepontis in 1978. Their book [loc. cit.] includes historical notes on several modifications of caliber.