Aspects of RG-spaces
DOI:
https://doi.org/10.4995/agt.2012.1637Keywords:
Almost Baire spaces, RG-spaces, Blumberg spaces, Almost resolvable spaces, Spaces of countable pseudocharacter, Prime zideal, P-space, Almost-P spaceAbstract
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.
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