An Upper Bound for the Weight of the Fine Uniformity

If (X,U) is a Hausdorff uniform space, we define the uniform weight w(X,U) as the smallest cardinal κ such that U has a basis of cardinality κ. An important topological cardinal of a Tychonoff space X is the number of cozero sets of X, which we denote as z(X). It is known that w(X,U)≤z(X×X) for every compatible uniformity U of X. We do not know if z(X×X) can be replaced by z(X). We concentrate ourselves in w(X,Un), where Un is the fine uniformity of X, i.e., the one having the family of normal covers as a basis. We establish upper bounds for w(X,Un) using the character and pseudocharacter in extensions of X×X or using the cardinal z(X). We also find some generalizations of the equivalence: w(X,Un)=ℵ0 if and only if X is metrizable and the set of non-isolated points of X is compact.