Some category bases which are equivalent to topologies (Q1124161)

A category base is a pair (X,\({\mathcal C})\) where \({\mathcal C}\) is a family of subsets of nonempty set X which satisfies certain general set theoretic conditions. This concept unites features of the topological theory of category with features of measure theory. Every topological space is a category base; and a general question is: which category bases are equivalent to topological spaces? This paper gives a sufficient condition for this to be true, and applies it to two examples. A special case is that (assuming CH) for any Hausdorff measure \(\mu\) on \({\mathbb{R}}^ n\), there exists a topology \({\mathcal T}\) on \({\mathbb{R}}^ n\) such that the \(\mu\)- measurable sets are precisely the sets with the Baire property with respect to \({\mathcal T}\).