For any topological space X we study the relation between the universal uniformity UX, the universal quasi-uniformity d40e852bb97916c3a4e67" title="Click to view the MathML source">qUX and the universal pre-uniformity pUX on X . For a pre-uniformity U on a set X and a word v in the two-letter alphabet 54d7884a45e2be7" title="Click to view the MathML source">{+,−} we define the verbal power Uv of U and study its boundedness numbers ℓ(Uv), , L(Uv) and . The boundedness numbers of (the Boolean operations over) the verbal powers of the canonical pre-uniformities pUX, d40e852bb97916c3a4e67" title="Click to view the MathML source">qUX and UX yield new cardinal characteristics ℓv(X), , Lv(X), , qℓv(X), 546b0df34f95b0cc198f134f">, 544167b49b33dc0568e1a" title="Click to view the MathML source">qLv(X), , 4d3917aa52a361452" title="Click to view the MathML source">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 ℓv, to classical cardinal topological invariants such as Lindelöf number ℓ, density d, and spread s . The simplest new verbal cardinal invariant is the foredensity 54f8c3" title="Click to view the MathML source">ℓ−(X) defined for a topological space X as the smallest cardinal κ such that for any neighborhood assignment (Ox)x∈X there is a subset A⊂X of cardinality |A|≤κ that meets each neighborhood 4d7d" title="Click to view the MathML source">Ox, 4d28" title="Click to view the MathML source">x∈X. It is clear that ℓ−(X)≤d(X)≤ℓ−(X)⋅χ(X). We shall prove that ℓ−(X)=d(X) if 54351c3f64f33de71c0" title="Click to view the MathML source">|X|<ℵω. On the other hand, for every singular cardinal κ (with κ≤22cf(κ)) we construct a (totally disconnected) T1-space X such that ℓ−(X)=cf(κ)<κ=|X|=d(X).