Verbal covering properties of topological spaces

Taras Banakh, Alex Ravsky

Published 2015-03-15Version 1

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 $\ell^-(X)=cf(\kappa)<\kappa=|X|=d(X)$.