# When is the weak topology generated by a family of functions Baire?

Suppose we are given a locally compact space $$X$$ with $$C_b(X)$$ denoting the continuous bounded complex or real functions on $$X$$.

Now, if $$A\subset C_b(X)$$ is given, I am trying to figure out when the weak topology induced by $$A$$ on $$X$$, i.e. the weakest topology on $$X$$ making functions in $$A$$ continuous, has the Baire property.

In particular, if $$A$$ separates the points of $$X$$ and thus induces a Hausdorff topology, what extra properties of $$A$$ may ensure local compactness, compactness? Does assuming that $$A$$ is a C*-algebra help?

I haven't been able to find any references on general weak topologies, so I would really appreciate any input or intuition.

• I think that the edit to the title is erroneous. – Douglas Somerset Jul 8 '19 at 22:38
• @DouglasSomerset It indeed is! – Merry Jul 9 '19 at 19:01
• If you assume that $A$ is a C$^*$-algebra then $X/R$ will be the Gelfand space of $A$, where $R$ is the equivalence relation on $X$ consisting of points not separated by $A$. So $X/R$ will be locally compact and Baire, and hence $X$ will be too in the weak topology. – Douglas Somerset Jul 9 '19 at 20:42
• I don't think assuming that $A$ is a C* algebra is essential actually, since a collection of functions generates the same topology as the collection of all possible C* combinations of them, and a collection generates the same topology as its closure in $C_b(X)$. – erz Jul 11 '19 at 9:15