# The Topologies generated by Borel sets.

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 $[a,b) = \bigcap_{n=1}^{\infty} (a-\frac{1}{n}, b)$.

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In the case of $\Bbb R$ you get the discrete topology, since each singleton is a $G_\delta$. This is true in any metric space, in fact. –  Brian M. Scott Sep 9 '13 at 20:55
ah, then also in any perfect space because there also every singleton is a $G_{\delta}$ set. –  Stefan Sep 9 '13 at 21:04
Any perfect $T_1$ space; there are non-$T_1$ spaces in which every closed set is a $G_\delta$. –  Brian M. Scott Sep 9 '13 at 21:06
You might get something interesting by looking at the topology generated by the effectively $G_\delta$ sets, the $\Pi^0_2$ ("lightface") sets. The topology generated by the $\Sigma^1_1$ sets ("lightface") is called the Gandy-Harrington topology and has been studied. –  Archimondain Sep 10 '13 at 11:49