About Homogeneous Spaces and the Baire Property in Remainders

Александър В. Архангелски, Митрофан М. Чобан, Екатерина П. Михайлова - В съобщението е продължено изследването на понятията o-хомогенно пространство, lo-хомогенно пространство, do-хомогенно пространство и co-хомогенно пространство. Показано е, че ако co-хомогенното пространство X съдържа Gδ -гъсто Московско подпространство, тогава X е Московско пространство.

About Topological Groups and the Baire Property in Remainders

Александър В. Архангелски, Митрофан М. Чобан, Екатерина П. Михайлова - Изследвани са прирасти със свойството на Бер на топологични групи.

Projectively Solid Sets and an N-dimensional Piccard's Theorem

We discuss functions f : X × Y → Z such that sets of the form f (A × B) have non-empty interiors provided that A and B are non-empty sets of second category and have the Baire property.

Boolean Rings that are Baire Spaces

∗ The present article was originally submitted for the second volume of Murcia Seminar on Functional Analysis (1989). Unfortunately it has been not possible to continue with Murcia Seminar publication anymore. For historical reasons the present vesion correspond with the original one.

Compactness in the First Baire Class and Baire-1 Operators

For a polish space M and a Banach space E let B1 (M, E) be the space of first Baire class functions from M to E, endowed with the pointwise weak topology. We study the compact subsets of B1 (M, E) and show that the fundamental results proved by Rosenthal, Bourgain, Fremlin, Talagrand and Godefroy, in case E = R, also hold true in the general case. For instance: a subset of B1 (M, E) is compact iff it is sequentially (resp. countably) compact, the convex hull of a compact bounded subset of B1 (M, E) is relatively compact, etc. We also show that our class includes Gulko compact. In the second part of the paper we examine under which conditions a bounded linear operator T : X ∗ → Y so that T |BX ∗ : (BX ∗ , w∗ ) → Y is a Baire-1 function, is a pointwise limit of a sequence (Tn ) of operators with T |BX ∗ : (BX ∗ , w∗ ) → (Y, · ) continuous for all n ∈ N. Our results in this case are connected with classical results of Choquet, Odell and Rosenthal.