Cardinality Bounds for Hausdorff SDL Spaces

Abstract

We establish the cardinal inequality \(|X|\leq 2^{t(X)Hψ(X)}\) for every Hausdorff SDL space \(X\), where \(t(X)\) and \(Hψ(X)\) denote the tightness and the Hausdorff pseudocharacter of \(X\), respectively. Since both invariants are bounded by \(χ(X)\), this yields \(|X|\leq 2^{χ(X)}\). As a consequence, every first-countable Hausdorff strongly cellular--Lindelöf space has cardinality at most the continuum. These results answer Questions~2.1 and~2.2 of Bella and Spadaro. An intermediate result is a uniform bounded-decomposition property for SDL spaces; in particular, their strict quasi--Lindelöf number satisfies \(\sqL(X)\leq t(X)\).