Weight of precompact subsets and tightness
In the paper under review, the authors study the weight and the tightness of a topological space \(X\), obtaining both well-known results and new interesting results in the setting of locally convex spaces (LCS). The authors first evaluate the weight of precompact subsets in uniform spaces with decreasing bases for their uniformity and then they apply the result to many LCS. In particular, they extend the main result of [\textit{B. Cascales} and \textit{J. Orihuela}, Proc. R. Soc. Edinb., Sect. A 103, 293-299 (1986; Zbl 0622.46005)] by showing that if \(E\) is the inductive limit of a sequence \((E_n)\) of LCS in \({\mathcal C}_m\), where \(m\) is an infinite cardinal number and \({\mathcal C}_m\) denotes the class of LCS \(E\) with character \(\chi(E)\leq m\), then \(w(K)\leq m\) for any precompact set \(K\subset E\); hence \(w(K)\leq \aleph_0\) for any precompact set \(K\) in an (LM)-space. They also obtain new results on countable tightness of LCSs. Indeed, it is proved that if \(E\) is the inductive limit of at most \(m\) locally convex spaces in \({\mathcal C}_m\), then \(t(E)\leq m\) and \(t(E,\sigma (E,E'))\leq m\). Then it turns out that \(t(E)\leq \aleph_0\) for every (LM)-space \(E\). Furthermore, the authors consider the large class \({\mathcal C}\) of LCSs introduced in [\textit{B. Cascales} and \textit{J. Orihuela}, Math. Z. 195, 365-381 (1987; Zbl 0604.46011)] and study when the spaces in \({\mathcal C}\) have countable tightness, obtaining: if \(E\) is an LCS in the class \({\mathcal C}\), then \((E,\sigma(E,E'))\) has countable tightness if and only if \((E,\sigma (E, E'))\) is real-compact; every quasi-barrelled LCS \(E\) in the class \({\mathcal C}\) has countable tightness. An example of a (DF) space which does not have countable tightness is given.
scientific article; zbMATH DE number 1797494
scientific article