2015/04/08 by Gotchev, Ivan S. · 1 citation
Mathematics · #54A25 (Primary) #54D20 (Secondary) #FOS: Mathematics #General Topology (math.GN) #math.GN #msc:54A25 #msc:54D20
paper · pdf · doi:10.48550/arxiv.1504.01785
12 pages
For a Urysohn space X we define the regular diagonal degree Δ(X) of X to be the minimal infinite cardinal κ such that X has a regular Gκ-diagonal i.e. there is a family (Uη:η<κ) of open neighborhoods of ΔX=\(x,x)∈ X2:x∈ X\ in X2 such that ΔX = \bigcapη<κ Uη. In this paper we show that if X is a Urysohn space then: (1) |X|≤ 2c(X)⋅Δ(X); (2) |X|≤ 2^Δ(X)⋅ 2wL(X); (3) |X|≤ wL(X)Δ(X)⋅χ(X); and (4) |X|≤ aL(X)Δ(X); where χ(X), c(X), wL(X) and aL(X) are respectively the character, the cellularity, the weak Lindelöf number and the almost Lindelöf number of X. The first inequality extends to the uncountable case Buzyakova's result that the cardinality of a ccc-space with a regular Gδ-diagonal does not exceed 2ω. It follows from (2) that every weakly Lindelöf space with a regular Gδ-diagonal has cardinality at most 22ω. Inequality (3) implies that when X is a space with a regular Gδ-diagonal then |X|≤ wL(X)χ(X). This improves significantly Bell, Ginsburg and Woods inequality |X|≤ 2χ(X)wL(X) for the class of normal spaces with regular Gδ-diagonals. In particular (3) shows that the cardinality of every first countable space with a regular Gδ-diagonal does not exceed wL(X)ω. For the class of spaces with regular Gδ-diagonals (4) improves Bella and Cammaroto inequality |X|≤ 2χ(X)⋅ aL(X), which is valid for all Urysohn spaces. Also, it follows from (4) that the cardinality of every space with a regular Gδ-diagonal does not exceed aL(X)ω.