On decomposition of a space into certain subsets and \({\text cl}\)- cardinality of a topological space
The idea is developed to estimate the power of a space \(X\) by means of the least cardinal number \(\kappa\) which is sufficient to decompose each subset of \(X\) into \(\kappa\) sets with certain property; see also the same authors' paper [ibid. 307-316 (1994; see the review below)]. The authors introduce the cardinal invariant \(\text{clard} (X)\) to be the least cardinal \(\kappa\) such that each subset of \(X\) is the union of a family of size at most \(\kappa\) consisting of closed subsets of \(X\). The main result of the paper says that if \(X\) is a \(T_ 1\)-space and \(w(X) \leq | X |\), then the power of \(X\) is smaller than the cofinality of \(2^{\text{clard} (X) + \ell (X)}\). In particular, under the assumption of the Generalized Luzin Hypothesis, \(| X | = \text{clard} (X) + \ell(X)\). It remains, however, the open question of whether the last equality is valid in ZFC.
scientific article; zbMATH DE number 600044
scientific article