2015/06/15 by Ivan S. Gotchev, Gotchev, Ivan S., Mikhail G. Tkachenko +3
Mathematics · #54A25 (Primary) #54D10 #54D20 (Secondary) #FOS: Mathematics #General Topology (math.GN) #math.GN #msc:54A25 #msc:54D10 #msc:54D20
paper · pdf · doi:10.48550/arxiv.1506.04665
14 pages. arXiv admin note: substantial text overlap with arXiv:1504.01785
arxiv created 2016/02/26 · arxiv updated 2016/02/29
We prove that, under CH, any space with a regular Gδ-diagonal and caliber ω1 is separable; a corollary of this result answers, under CH, a question of Buzyakova. For any Urysohn space X, we establish the inequality |X|≤ wL(X)^sΔ2(X)⋅dot(X) which represents a generalization of a theorem of Basile, Bella, and Ridderbos. We also show that if X is a Hausdorff space, then |X|≤(πχ(X)⋅ d(X))ot(X)⋅ψc(X); this result implies Šapirovski\uı's inequality |X|≤πχ(X)c(X)⋅ψ(X) which only holds for regular spaces. It is also proved that |X|≤ πχ(X)ot(X)⋅ψc(X)⋅ aLc(X) for any Hausdorff space X; this gives one more generalization of the famous Arhangel^′skii's inequality |X|≤ 2χ(X)⋅ L(X).