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 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 $\ell^-(X)=cf(\kappa)<\kappa=|X|=d(X)$.Comment: 20 pages, many diagram

Some remarks on topologized groups

“…called the U-neighborhood of A. We refer the reader to [6,7] for more information on pre-uniform spaces. Definition 2.1.…”

Pinning down versus density

“…The pinning down number has been recently introduced in [2] under the name "foredensity" and it was denoted there by ℓ − (X). The following two interesting results concerning the pinning down number were also established in [2]:…”

“…• [2,Corollary 5.4] If κ is any singular cardinal then there is a T 1 semitopological group X such that pd(X) = cf(κ) < κ = d(X) = |X| = ∆(X).…”

“…• [2,Problem 5.5] Is there a ZFC example of a Hausdorff space X with pd(X) < d(X)? • [2,Problem 5.6] Is it consistent to have a regular space X with pd(X) < d(X)?…”

“…The examples that Banakh and Ravsky constructed in the proof of [2,Corollary 5.4], as well as the examples we first constructed in our proof of theorem 1.2 were both neat and of singular cardinality. Hence it was natural for us to raise the question if witnesses for pd(X) < d(X) that are both neat and of regular cardinality could also be found.…”