Verbal covering properties of topological spaces

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 $\mathrm{\ell }({\mathcal{U}}^v)$ and $\bar{\mathrm{\ell }}({\mathcal{U}}^v)$. 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 ${\mathrm{\ell }}^v(X)$, ${\bar{\mathrm{\ell }}}^v(X)$, $q{\mathrm{\ell }}^v(X)$, $q{\bar{\mathrm{\ell }}}^v(X)$, $u\mathrm{\ell }(X)$ 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 ${\mathrm{\ell }}^v$, ${\bar{\mathrm{\ell }}}^v$ to classical cardinal topological invariants such as Lindel"of number $\mathrm{\ell }$, density $d$, and spread $s$. The simplest new verbal cardinal invariant is the foredensity ${\mathrm{\ell }}^{-}(X)$ 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 $\mathrm{\mid }A\mathrm{\mid }\le \kappa$ that meets each neighborhood $O_x$, $x\in X$. It is clear that $\ell^-(X)\le d(X)\le \ell^-(X)\cdot \chi(X)$\mathrm{.}Weshallprovethat$\ell^-(X)=d(X)$if$|X|<\aleph_\omega$\mathrm{.}Ontheotherhand,foreverysingularcardinal$\kappa$(with$\kappa\le 2^{2^{cf(\kappa)}}$)weconstructa(totallydisconnected)$T_1$-space$X$suchthat$\ell^-(X)=cf(\kappa)<\kappa=|X|=d(X)$.