On decomposition of a space into certain subsets and cl-cardinality of a topological space

AbstractBy cl-cardinality of a space X we call the cardinal clard (X) =min{τ:each subset (of X) is a union of ⩽ τ closed in X subspaces}. Some relations between cl-cardinality and cardinality of a space are established. Among them |X| < cf(2clard(X)·l(X)⩽2clard(X)·l(X) and under GLH, |X| = clard(X)·l(X) for each weakly additional T1-space (in particular, for each T2-space of point-countable type). (The equality is independent of ZFC; GLH: “2τ < 2τ+ for every cardinal τ”.) Besides, under GLH, |X|⩽clard(X)l(X) for every T1-space X