2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/126529All spaces below are Tychonov. We define the projective pi-character p(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 pi-base whose order is at most p(X), that is every point in X is contained in at most p(X)-many members of the pi-base. Since p(X) is at most t(X) for compact X, this provides a significant generalization of a celebrated result of Shapirovskii.7 pagesGeneral Topology54A25text