The Topologies generated by Borel sets.

Question

If I have a topological space $(X, \tau)$, then we can consider the Borel hierarchy on it. Now the sets from a Borel class itself could be taken as generating a topology. Is something known or what could be said about the properties of these topologies? In particular for the Topology generated by all $G_{\delta}$ sets? For example for the Standardtopology on $\mathbb R$ you get, by using $G_{\delta}$ sets, a topology which is a refinement of the Sorgenfrey line because $[a,b) = \bigcap_{n=1}^{\infty} (a-\frac{1}{n}, b)$.

Answer 1

If you want to keep the topology Polish, then you can add countably many Borel sets as clopen sets. Details are in Chapter 13 of the book of A.S. Kechris "Classical Descriptive Set Theory".

As mentioned in one of the comments, you may add countably many analytic sets. Then the topology may not be normal anymore, but according to Theorem 25.18 in the above mentioned book, the space remains strong Choquet, that is a game-theoretic version of the Baire category theorem still holds.

Adding uncountably many Borel sets as clopen sets opens the doors to all sort of interesting examples in the general topology, such as the Michael line (for a discussion of these examples, you may consult R. Engelking's book "General Topology", Chapter 5, Examples 5.1.22, 5.1.32 or a blog post by Dan Ma).