2 8 M ar 2 00 7 PROJECTIVE π - CHARACTER BOUNDS THE ORDER OF A π - BASE

All spaces below are Tychonov. We define the pro-jective π-character p πχ(X) of a space X as the supremum of the values πχ(Y) where Y ranges over all continuous images of X. Our main result says that every space X has a π-base whose order is ≤ p πχ(X), that is every point in X is contained in at most p πχ(X)-many members of the π-base. Since p πχ(X) ≤ t(X) for compact X, this provides a significant generalization of a celebrated result of Shapirovskii. Arhangel'skii has recently introduced in [1] the concept of a space of countable projective π-character and noticed that any compact space of countable tightness has countable projective π-character. Then he showed that a compact space of countable projective π-character having ω 1 as a caliber is separable, thereby strengthening Shapirovskii's analogous result for countably tight compacta. Note that Shapirovskii's theorem is a trivial corollary of his more general result establishing that any countably tight compactum has a point-countable π-base, or more generally: any compactum X has a π-base of order at most t(X), see [2] or [4]. In this paper we use the general concept of projective π-character to give the following significant generalization of this stronger result of Shapirovskii: Any Tychonov space has a π-base of order at most the projective π-character of the space. Not only is this result stronger for compacta, by replacing tightness with projective π-character that is smaller, but somewhat surprisingly it extends to all Tychonov spaces. Let ϕ be any cardinal function defined on a class C of topological spaces. We define the projective version p ϕ of ϕ as follows. For any X ∈ C we let p ϕ(X) be the the supremum of the values ϕ(Y) where Y ranges over all continuous images of X in C. In particular, we shall consider the case in which ϕ = πχ, the π-character defined on the class