Baire category and 𝐵ᵣ-spaces

Noll, Dominikus

doi:10.1090/S0002-9939-1987-0911031-0

Baire category and $B_ r$-spaces
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by Dominikus Noll PDF

Proc. Amer. Math. Soc. 101 (1987), 671-678 Request permission

Abstract:

A topological space satisfying the open mapping theorem is called a ${B_r}$-space. We investigate the question whether completely regular ${B_r}$-spaces must be Baire spaces. The answer we obtain is twofold and surprising. On the one hand there exist first category completely regular ${B_r}$-spaces. Examples are provided in the class of Lindelöf $P$-spaces. On the other hand, we obtain a partial positive answer to our question. We prove that every suborderable metrizable ${B_r}$-space is in fact a Baire space. We conjecture that this is true for metrizable ${B_r}$-spaces in general. Our paper is completed by some applications. For instance, we establish the existence of a metrizable ${B_r}$-space $E$ whose square $E \times E$ is no longer a ${B_r}$-space.

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