Schwartz writes in his book "Radon Measures on Arbitrary Topological Spaces", p. 110: "But even a finite union of polish spaces need not be polish". The same statement can be also found here.
How should this statement be interpreted? If I have a Polish space $X$ and a finite union of Polish subspaces $X_i \subseteq X$ then all of them are $G_\delta$-subsets and their (finite) union is again a $G_\delta$-subset of $X$ thus a Polish space.
Should this statement be interpreted that the given Polish spaces $X_i$ are not necessarily contained in a joint Polish space $X$? But what topology do we impose on the union? It may not be a disjoint union (which would be again Polish with the disjoint union topology).