For any topological space X we study the relation between the universal uniformity UX, the universal quasi-uniformity 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 {+,−} we define the verbal power Uv of U and study its boundedness numbers 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 pUX, qUX and UX yield new cardinal characteristics ℓv(X), ℓ¯v(X), Lv(X), L¯v(X), qℓv(X), qℓ¯v(X), qLv(X), qL¯v(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 ℓv, 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) 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 Ox, x∈X. It is clear that ℓ−(X)≤d(X)≤ℓ−(X)⋅χ(X). We shall prove that ℓ−(X)=d(X) if |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).