The Non-urysohn Number of a Topological Space

We call a nonempty subset A of a topological space X finitely non-Urysohn if for every nonempty finite subset F of A and every family {Ux : x ∈ F} of open neighborhoods Ux of x ∈ F , ∩{cl(Ux) : x ∈ F} 6 = ∅ and we define the non-Urysohn number of X as follows: nu(X) := 1 + sup{|A| : A is a finitely non-Urysohn subset of X}. Then for any topological space X and any subset A of X we prove the following inequalities: (i) |clθ(A)| ≤ |A| · nu(X); (ii) |[A]θ| ≤ (|A| · nu(X)); (iii) |X| ≤ nu(X)θ; and (iv) |X| ≤ nu(X). In 1979, A. V. Arhangel′skĭı asked if the inequality |X| ≤ 2c was true for every Hausdorff spaceX. It follows from the third inequality that the answer of this question is in the affirmative for all spaces with nu(X) not greater than the cardinality of the continuum. We also give a simple example of a Hausdorff space X such that |clθ(A)| > |A|U(X) and |clθ(A)| > (|A| · U(X)), where U(X) is the Urysohn number of X, recently introduced by Bonanzinga, Cammaroto and Matveev. This example shows that in (i) and (ii) above, nu(X) cannot be replaced by U(X) and answers some questions posed by Bella and Cammaroto (1988), Bonanzinga, Cammaroto and Matveev (2011), and Bonanzinga and Pansera (2012).