Here is an example by Roman Sikorski:

$$(H\setminus\{\mathbf 0\})\,^\mathbb N$$

in $\ H=_{top} H\,^\mathbb N,\ $where $\ H\ $ is the countably-dimensional Hilbert space (you can have the Hilbert cube instead). Of course any metric space would do, but this one is universal for $G_\delta$-sets in the Sikorski's sense. This is just one of the double transfinite sequence of universal sets for all Borel classes. Sikorski called Borel classes additive and productive, so that--for instance--$G_\delta$ would be productive of class 1.