Aspects of RG-spaces

[EN] A Tychonoff space X which satisfies the property that G(X) = C(Xδ) is called an RG-space, where G(X) is the minimal regular ring extension of C(X) inside F(X), the ring of all functions from X to R, and Xδ is the topology on X generated by its Gδ-sets. We correct an error tha twe found in the proof of and show that RG-spaces must satisfy a finite dimensional condition.

We also introduce a new class of topological spaces which we call almost k-Baire spaces. The class of almost Baire spaces is a particular instance. We show that every RG-space is an almost Baire space but not necessarily a Baire space. However RG-spaces of countable pseudocharacter must be Baire and, furthermore, their dense sets have dense interiors.