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Let S be a subset of the reals such that S∩[a,b] and S^c∩[a,b] cannot be written as a countable union of closed sets for any a<b. This can be done (this explicit example of a non-Borel set achieves this). Let ℚ be the rationals. Then, A=(Sxℚ)U(S^cxℚ^c) and B=(Sxℚ^c)U(S^cxℚ) should do it.