Baire category and 𝐵ᵣ-spaces

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