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