Open subsets of perfect Polish spaces Is it true that every non-empty open subset of a perfect Polish space
is uncountable?
It is true that the space itself is uncountable but I was not able to show 
that every non-empty open subset is uncountable.

$G_\delta$-sets in a completely metrizable space are completely metrizable, so the same is certainly true of open sets. An open subset of a separable metrizable space is also separable and metrizable. Finally, a non-empty open subset of a perfect space is perfect: an isolated point in it would be isolated in the whole space. Thus, every non-empty subset of a perfect Polish space is a perfect Polish space and therefore has cardinality $2^\omega=\mathfrak{c}$.
