Baire category and ᵣ-spaces

A topological space satisfying the open mapping theorem is called a Br-space. We investigate the question whether completely regular Br-spaces must be Baire spaces. The answer we obtain is twofold and surprising. On the one hand there exist first category completely regular Br-spaces. Examples are provided in the class of Lindelof P-spaces. On the other hand, we obtain a partial positive answer to our question. We prove that every suborderable metrizable Br-space is in fact a Baire space. We conjecture that this is true for metrizable Br-spaces in general. Our paper is completed by some applications. For instance, we establish the existence of a metrizable Br-space E whose square E x E is no longer a Br-space. Introduction. A Hausdorff topological space E is called a Br-space (resp. a Bspace) if every continuous, nearly open bijection (resp. surjection) f from E onto an arbitrary Hausdorff topological space F is open. Every locally compact Hausdorff space is a B-space and every B-space is a BrT space. Every tech complete space is known to be a Br-space (see [BP]). Br-spaces have been investigated in several papers; see [We, BP, N1, N2]. In our paper [N2] we have examined the problem of invariance of the class of Brspaces under the operation of taking finite sums. We have obtained the following result. Let E be a completely regular Br-space. Then the following statements are equivalent: (1) E is a Baire space. (2) Whenever F is a tech complete space, then the topological sum E E F is again a Br-space. We left open in [N2] the question whether completely regular Br-spaces are Baire-spaces in general, i.e., whether statement (2) above is true for arbitrary completely regular Br-spaces E. In fact, one is tempted to conjecture that the answer to this question is in the positive if one takes into account the following facts: (a) Given an arbitrary Br-space E, the sum E eDE is again a Br-space (see [N2, Theorem 1]), and the same is true for any sum EeL, where L is an arbitrary locally compact Hausdorff space. So why should not E E F be a Br-space for arbitrary tech complete spaces F? (b) There is a partial positive answer to our question in the class of strongly zero-dimensional metrizable Br-spaces (which we shall prove in the present paper) stating that every strongly zero-dimensional metrizable Br-space is in fact a Baire space. Received by the editors December 31, 1985 and, in revised form, August 11, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 46A30, 02G20, 54C10. (D1987 American Mathematical Society 0002-9939/87 $1.00 + $.25 per page