BOUNDING THE MINIMUM ORDER OF A π-BASE

Abstract

Let X be a topological space. A family ℬ of nonempty open sets in X is called a π-base of X if for each open set U in X there exists B∈ℬ such that B⊂U. The order of a π-base ℬ at a point x is the cardinality of the family ℬ_x ={B∈ℬ:x∈B} and the order of the π-base ℬ is the supremum of the orders of ℬ at each point x∈X. A classical theorem of Shapirovskiĭ [‘Special types of embeddings in Tychonoff cubes’, in: Subspaces of Σ-Products and Cardinal Invariants, Topology, Coll. Math. Soc. J. Bolyai, 23 (North-Holland, Amsterdam, 1980), pp. 1055–1086; ‘Cardinal invariants in compact Hausdorff spaces’, Amer. Math. Soc. Transl.134 (1987), 93–118] establishes that the minimum order of a π-base is bounded by the tightness of the space when the space is compact. Since then, there have been many attempts at improving the result. Finally, in [‘The projective π-character bounds the order of a π-base’, Proc. Amer. Math. Soc.136 (2008), 2979–2984], Juhász and Szentmiklóssy proved that the minimum order of a π-base is bounded by the ‘projective π-character’ of the space for any topological space (not only for compact spaces), improving Shapirovskiĭ’s theorem. The projective π-character is in some sense an ‘external’ cardinal function. Our purpose in this paper is, on the one hand, to give bounds of the projective π-character using ‘internal’ topological properties of the subspaces on compact spaces. On the other hand, we give a bound on the minimum order of a π-base using other cardinal functions in the frame of general topological spaces. Open questions are posed.