The Projective Π-character Bounds the Order of a Π-base

All spaces below are Tychonov. We define the projective πcharacter p πχ(X) of a space X as the supremum of the values πχ(Y ) where Y ranges over all (Tychonov) 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 is a significant generalization of a celebrated result of Shapirovskii. Let us start by recalling a few definitions and basic facts. A π-base B of a space X (resp. a local π-base at a point x ∈ X) is a family of non-empty open sets such that every non-empty open set (resp. every neighbourhood of x) includes some member of B. The π-weight π(X) of X is the smallest infinite cardinal such that X has a π-base of at most that cardinality. The π-character πχ(x,X) of x in X is the smallest cardinality of a local π-base at x ∈ X, and πχ(X) = sup{πχ(x,X) : x ∈ X} is the π-character of the space X. Finally, the local tightness at x ∈ X is the smallest cardinal κ such that if x belongs to the closure A of a set A, then there is B ⊂ A with |B| ≤ κ and x ∈ B; moreover t(X) = sup{t(x,X) : x ∈ X} is the tightness of the space X. Shapirovskii proved the following two important results concerning these cardinal functions for compacta: If X is compact, then πχ(X) ≤ t(X). Moreover X has a π-base B of order ≤ t(X); i.e. every point of X is contained in at most t(X)-many members of B. (A trivial consequence is that if t(X) is a caliber of X—i.e. among t(X)-many open sets there are always t(X)-many with non-empty intersection— then X has a π-base of cardinality at most t(X).) The first result was proved in [5], and alternative proofs were given in [1] and [3]. The second result first appeared in [6] and then in [8]. A very short and elegant new proof (using a variant of Shapirovskii’s “algebraic” approach to free sequences) was presented in [9]. Arhangel’skii has recently introduced in [2] the concept of a space of countable projective π-character and noticed that any compact space of countable tightness Received by the editors March 28, 2007, and, in revised form, June 17, 2007. 2000 Mathematics Subject Classification. Primary 54A25, 54C10, 54D70.