2015/04/08 by Ivan S. Gotchev, Gotchev, Ivan S.
Mathematics · #FOS: Mathematics #General Topology (math.GN) #math.GN
paper · pdf · doi:10.48550/arxiv.1504.01790
10 pages
arxiv created 2015/04/08 · arxiv updated 2015/04/09
A non-empty subset A of a topological space X is called finitely non-Hausdorff if for every non-empty finite subset F of A and every family \Ux:x∈ F\ of open neighborhoods Ux of x∈ F, ∩\Ux:x∈ F\≠∅ and the non-Hausdorff number nh(X) of X is defined as follows: nh(X):=1+sup\|A|:A⊂ X is finitely non-Hausdorff\. Clearly, if X is a Hausdorff space then nh(X)=2. We define the non-Urysohn number of X with respect to the singletons, nus(X), as follows: nus(X):=1+sup\clθ(\x\):x∈ X\. In 1967 Hajnal and Juhász proved that if X is a Hausdorff space then: (1) |X|≤ 2c(X)χ(X); and (2) |X|≤ 2^2s(X); where c(X) is the cellularity, χ(X) is the character and s(X) is the spread of X. In this paper we generalize (1) by showing that if X is a topological space then |X|≤ nh(X)c(X)χ(X). Immediate corollary of this result is that (1) holds true for every space X for which nh(X)≤ 2ω (and even for spaces with nh(X)≤ 2c(X)χ(X)). This gives an affirmative answer to a question posed by M. Bonanzinga in 2013. A simple example of a T1, first countable, ccc-space X is given such that |X|>2ω and |X|=nh(X)ω=nh(X). This example shows that the upper bound in our inequality is exact and that nh(X) cannot be omitted (in particular, nh(X) cannot always be replaced by 2 even for T1-spaces). In this paper we also generalize (2) by showing that if X is a T1-space then |X|≤ 2^nus(X)⋅ 2s(X). It follows from our result that (2) is true for every T1-space for which nus(X)≤ 2s(X). A simple example shows that the presence of the cardinal function nus(X) in our inequality is essential.