The non-Urysohn number of a topological space

TL;DR

This paper introduces the non-Urysohn number of a topological space, establishes inequalities relating it to other topological invariants, and explores its implications for classical questions in topology, including counterexamples and bounds.

Contribution

It defines the non-Urysohn number, derives new inequalities involving it, and provides examples showing its distinctness from the Urysohn number, advancing understanding of topological cardinal invariants.

Findings

Established inequalities relating non-Urysohn number to closure and cardinality.

Provided examples demonstrating the non-Urysohn number's independence from the Urysohn number.

Confirmed some classical inequalities for spaces with bounded non-Urysohn number.

Abstract

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 $\{U_x:x\in F\}$ of open neighborhoods $U_x$ of $x\in F$, $\cap\{\mathrm{cl}(U_x):x\in F\}\ne\emptyset$ 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: (1) $|\mathrm{cl}_\theta(A)|\le |A|^{\kappa(X)}\cdot nu(X)$, (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 $|X|\le 2^{\chi(X)wL_c(X)}$ was true for every Hausdorff space $X$. 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 $|\mathrm{cl}_\theta(A)|>|A|^{\chi(X)}U(X)$ and $|\mathrm{cl}_\theta(A)|>(|A|\cdot U(X))^{\chi(X)}$, 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).