Consider a measurable space with a finite vector measure. This measure defines a mapping of the σ-field into a Euclidean space. According to Lyapunov’s convexity theorem, the range of this mapping is compact and, if the measure is atomless, this range is convex. Similar ranges are also defined for measurable subsets of the space. We show that the union of the ranges of all subsets having the sa...

For any space X, denote by dis(X) the smallest (infinite) cardinal κ such that κ many discrete subspaces are needed to cover X. It is easy to see that if X is any crowded (i.e. dense-in-itself) compactum then dis(X) m, where m denotes the additivity of the meager ideal on the reals. It is a natural, and apparently quite difficult, question whether in this inequality m could be replaced by c. He...

The weak tightness wt(X), introduced in [6], has the property wt(X)≤t(X). It was shown [4] that if X is a homogeneous compactum then |X|≤2wt(X)πχ(X). We introduce almost at(X) with wt(X)≤at(X)≤t(X) and show power |X|≤2at(X)πχ(X). This improves result of Arhangel′skiĭ, van Mill, Ridderbos [2] |X|≤2t(X) for gives partial answer to question [4]. In addition, Hausdorff space we |X|≤2pwcL(X)wt(X)πχ(...

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 signifi...

A branchpoint of a compactum X is a point which is the vertex of a simple triod in X. A surjective map /: X -» Y is said to cover the branchpoints of Y if each branchpoint in Y is the image of some branchpoint in X. If every map in a class % of maps on a class of compacta & covers the branchpoints of its image, then it is said that the branchpoint covering property holds for ff on 0. According ...