Verbal covering properties of topological spaces

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

20 pages, many diagrams

References in corpus (2)

• On critical cardinalities related to $Q$-sets
• On the submetrizability number and $i$-weight of quasi-uniform spaces and paratopological groups

Cited by in corpus (1)

• On the submetrizability number and $i$-weight of quasi-uniform spaces and paratopological groups