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 , U n ) , where U n 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 , U n ) 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 , U n ) = ℵ 0 if and only if X is metrizable and the set of non-isolated points of X is compact.