2021/09/23 by Istvan Juhasz, Juhasz, Istvan, Lajos Soukup +3
Mathematics · #54A25 #54A35 #FOS: Mathematics #General Topology (math.GN) #math.GN #msc:54A25 #msc:54A35
paper · pdf · doi:10.48550/arxiv.2109.11432
4 pages
arxiv created 2021/09/23 · arxiv updated 2021/09/24
The main result of this note is the following theorem. "If X is any Hausdorff space with κ= \widehatF(X) ⋅ \widehatμ(X) then L(X< κ) ≤ \varrho(κ)". Here \widehatF(X) is the smallest cardinal φ so that |S| < φ for any set S that is free in X and \widehatμ(X) is the smallest cardinal μ so that, for every set S that is free in X, any open cover of S has a subcover of size < μ. Moreover, X< κ is the G< κ-modification of X and \varrho(κ) = min \\varrho : \varrho < κ = \varrho\. As a corollary we obtain that if X is a linearly Lindelöf regular space of countable tightness then L(Xδ) ≤ \mathfrakc, provided that \mathfrakc = 2^< \mathfrakc. This yields a consistent affirmative answer to a question of Angelo Bella.