A $G_\delta$-set is obtained from a countable intersection of open sets. Do we have any results about the intersection of uncountably many open sets? Or does it even make sense?
The $G_\delta$-sets arise naturally from considering the $\sigma$-algebra generated by the usual topology on the real line and I assume that this is how the $G_\delta$-sets are motivated. But what if we look at larger structures (meaning with cardinalities larger than the continuum) equipped with a topology that is, for example, not second-countable and so any basis is also necessarily large? The $G_\delta$-sets would form a smaller family compared to the family of sets formed by the intersections of say $\aleph_1$ open sets each, wouldn't they?
In any case, please enlighten me on the subject. I am unfamiliar.
In any Hausdorff topological space (in particular in a metric space) any subset is a union of singleton sets (which are closed). Hence, by taking complements, every subset is an intersection of open sets.