I have a Polish space $X$ and a subset $A \subset X$.

I know that $A$ is completely metrizable (in its induced topology) if and only if $A$ is a $G_\delta$-set in $X$.

This means: If I want to show that $A$ *is* completly metrizable then it suffices to find a sequence $U_1,U_2,\ldots$ of open sets such that $A=\bigcap_j U_j$.

But what if I want to show that $A$ is *not* completely metrizable? Are there any criteria I can use?
The only criterium I know is that if $A$ is not Baire then it is not completely metrizable (This shows that $\mathbb Q$ is not $G_\delta$ in $\mathbb R$) but this does not help if my set $A$ is a Baire space. Also there may be cases where it is not easy to decide wether $A$ is Baire.

I hope that something can be said in this generality... Thanks for your help in advance!