Verbal covering properties of topological spaces

For any topological space X   we study the relation between the universal uniformity UXUX, the universal quasi-uniformity qUXqUX and the universal pre-uniformity pUXpUX on X  . For a pre-uniformity UU on a set X and a word v   in the two-letter alphabet {+,−}{+,−} we define the verbal power UvUv of UU and study its boundedness numbers ℓ(Uv)ℓ(Uv), ℓ¯(Uv), L(Uv)L(Uv) and L¯(Uv). The boundedness numbers of (the Boolean operations over) the verbal powers of the canonical pre-uniformities pUXpUX, qUXqUX and UXUX yield new cardinal characteristics ℓv(X)ℓv(X), ℓ¯v(X), Lv(X)Lv(X), L¯v(X), qℓv(X)qℓv(X), qℓ¯v(X), qLv(X)qLv(X), qL¯v(X), uℓ(X)uℓ(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 ℓvℓv, ℓ¯v to classical cardinal topological invariants such as Lindelöf number ℓ, density d, and spread s  . The simplest new verbal cardinal invariant is the foredensity ℓ−(X)ℓ−(X) defined for a topological space X as the smallest cardinal κ   such that for any neighborhood assignment (Ox)x∈X(Ox)x∈X there is a subset A⊂XA⊂X of cardinality |A|≤κ|A|≤κ that meets each neighborhood OxOx, x∈Xx∈X. It is clear that ℓ−(X)≤d(X)≤ℓ−(X)⋅χ(X)ℓ−(X)≤d(X)≤ℓ−(X)⋅χ(X). We shall prove that ℓ−(X)=d(X)ℓ−(X)=d(X) if |X|<ℵω|X|<ℵω. On the other hand, for every singular cardinal κ   (with κ≤22cf(κ)κ≤22cf(κ)) we construct a (totally disconnected) T1T1-space X   such that ℓ−(X)=cf(κ)<κ=|X|=d(X)ℓ−(X)=cf(κ)<κ=|X|=d(X).