Sums and products ofBrspaces

Abstract

In reminiscence of Ptak's open mapping theorem, a topological space satisfying the open mapping theorem is called a Br space. This paper is devoted to the study of sums and products of Br spaces in the category of topological spaces. We prove that, in general, sums and products of even two Br spaces need no longer be Br. On the other hand, for any Br space E the sum E θ E is again a Br space. Moreover, since Cech complete spaces are known to be Br, we ask whether a sum E θ F is Br provided that E is Cech complete and F is Br. It turns out that, at least in the framework of complete regularity, the answer to this question is in the positive if and only if F is a Baire space. Introduction. The notion of Br spaces has first been introduced by T. Husain in the categories of locally convex vector spaces (see (HuJ) and topological groups (see (Hu2)). Originally, it goes back to V. Ptak's open mapping theorem (see (Kό), p. 35 ff). A Hausdorff topological space E is called a Br space if every continuous, nearly open bijection / from E onto any Hausdorff space F is in fact open. For a survey of the classical theory of Br spaces we refer the reader to Kόthe's book (Kό), where the locally convex case is treated. The linear topological case is investigated in the lecture notes (AEK). Br groups are considered by several authors. See for instance (Hu2), (Ba), (Gr), (Su). In a purely topological context, Br spaces have first been investigated by Weston in (We), although the term Br space is not used there. Translated into the Br terminology, Weston proved that every completely metrizable topological space is a Br space. His result has been generalized by Byczkowski and Pol in (BP), who proved that every Cech complete topological space is a Br space. In (No) we have further generalized this result proving that every Hausdorff, semi-regular topological space densely containing some Cech complete sub space is in fact a Br space. We just mention another generalization of Byczkowski and Pols' result into a somewhat different direction by Wilhelm (cf. (Wi)). In the present paper we examine the invariance of the class of Br spaces under topological sums and products. It turns out that, in general, the sum of even two Br spaces need not be a Br space. A counterexample is given in §2. In §1 we obtain a positive result stating that the sum E θ E