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 ${U_x:x\in F}$ofopenneighborhoods$U_x$of$x\in F$,$\cap{$\mathrm{c}\mathrm{l}$(U_x):x\in F}\ne\emptyset$andwedefinethenon-Urysohnnumberof$X$asfollows:$nu(X):=1+\sup{|A|:A$isafinitelynon-Urysohnsubsetof$X}$. Then for any topological space $X$ and any subset $A$ of $X$ we prove the following inequalities: (1) $|$\mathrm{c}\mathrm{l}$\theta(A)|\le |A|^{\kappa(X)}\cdot nu(X)$,(2)$|[A]\theta|\le (|A|\cdot nu(X))^{\kappa(X)}$,(3)$|X|\le nu(X)^{\kappa(X)sL_\theta(X)}$,and(4)$|X|\le nu(X)^{\kappa(X)aL(X)}$. In 1979, A. V. Arhangelskii asked if the inequality $|X|\le 2^{\chi(X)wL_c(X)}$wastrueforeveryHausdorffspace$X$. 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 $\mathrm{\mid }{\mathrm{c}\mathrm{l}}_{\theta }(A)\mathrm{\mid }>\mathrm{\mid }A{\mathrm{\mid }}^{\chi (X)}U(X)$ and $\mathrm{\mid }{\mathrm{c}\mathrm{l}}_{\theta }(A)\mathrm{\mid }>(\mathrm{\mid }A\mathrm{\mid }\cdot U(X){)}^{\chi (X)}$, where $U(X)$ is the Urysohn number of $X$, recently introduced by Bonanzinga, Cammaroto and Matveev. This example shows that in (1) and (2) 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).