Verbal covering properties of topological spaces

TL;DR

This paper introduces new cardinal invariants based on verbal powers of universal quasi-uniformities in topological spaces, linking them to classical invariants and exploring their properties and relationships.

Contribution

It defines and studies new verbal cardinal invariants of topological spaces, generalizing known invariants and establishing their relations to classical topological properties.

Findings

Introduced verbal cardinal invariants such as l^v, l^v, ql^v, and ul.

Showed l^-(X) equals the density d(X) for spaces with size less than f_f_f.

Constructed spaces demonstrating the bounds and differences of these invariants for singular cardinals.

Abstract

For any topological space $X$ we study the relation between the universal uniformity $\mathcal U_X$, the universal quasi-uniformity $q\mathcal U_X$ and the universal pre-uniformity $p\mathcal U_X$ on $X$. For a pre-uniformity $\mathcal U$ on a set $X$ and a word $v$ in the two-letter alphabet $\{+,-\}$ we define the verbal power $\mathcal U^v$ of $\mathcal U$ and study its boundedness numbers $\ell(\mathcal U^v)$ and $\bar \ell(\mathcal U^v)$. The boundedness numbers of the (Boolean operations over) the verbal powers of the canonical pre-uniformities $p\mathcal U_X$, $q\mathcal U_X$ and $\mathcal U_X$ yield new cardinal characteristics $\ell^v(X)$, $\bar \ell^v(X)$, $q\ell^v(X)$, $q\bar \ell^v(X)$, $u\ell(X)$ of a topological space $X$, which generalize all known cardinal topological invariants related to (star)-covering properties. We study the relation of the new cardinal invariants $\ell^v$, $\bar \ell^v$ to classical cardinal topological invariants such as Lindel\"of number $\ell$, density $d$, and spread $s$. The simplest new verbal cardinal invariant is the foredensity $\ell^-(X)$ defined for a topological space $X$ as the smallest cardinal $\kappa$ such that for any neighborhood assignment $(O_x)_{x\in X}$ there is a subset $A\subset X$ of cardinality $|A|\le\kappa$ that meets each neighborhood $O_x$, $x\in X$. It is clear that $\ell^-(X)\le d(X)\le \ell^-(X)\cdot \chi(X)$. We shall prove that $\ell^-(X)=d(X)$ if $|X|<\aleph_\omega$. On the other hand, for every singular cardinal $\kappa$ (with $\kappa\le 2^{2^{cf(\kappa)}}$) we construct a (totally disconnected) $T_1$-space $X$ such that $\ell^-(X)=cf(\kappa)<\kappa=|X|=d(X)$.