2 added 23 characters in body

Given a family of sets $G\subset P(X)$, can one characterize by "closure properties" alone whether or not $G$ arises as the family of all $G_\delta$ for some topology on $X$? some Polish space topology on $X$?

Of course I'm not interested in the trivial answer, namely: if there exists a system of sets $T$ closed under unions and finite intersections such that every $A\in G$ is a countable intersection of sets in $T$.

I don't feel sure about the best way to bar this non-answer! Feel free to improve my question if you have an answer.

I suppose I'm looking for something like this: (positive) statements with only universal quantifiers over combinatorially-restricted families of sets in $G$. Combinatorially restricted can allows for restrictions on the cardinality of the family, disjointness, nestedness, etc.

This is the easiest question of this type where I don't know the answer. So how about arbitrary levels of the Borel hierarchy?

1

# Closure properties of familes of $G_\delta$ sets.

Given a family of sets $G\subset P(X)$, can one characterize by "closure properties" alone whether or not $G$ arises as the family of all $G_\delta$ for some Polish space topology on $X$?

Of course I'm not interested in the trivial answer, namely: if there exists a system of sets $T$ closed under unions and finite intersections such that every $A\in G$ is a countable intersection of sets in $T$.

I don't feel sure about the best way to bar this non-answer! Feel free to improve my question if you have an answer.

I suppose I'm looking for something like this: (positive) statements with only universal quantifiers over combinatorially-restricted families of sets in $G$. Combinatorially restricted can allows for restrictions on the cardinality of the family, disjointness, nestedness, etc.

This is the easiest question of this type where I don't know the answer. So how about arbitrary levels of the Borel hierarchy?