It is shown that theσ-idealU
0 of closed sets of extended uniqueness inT is hereditarily non-Borel, i.e. every “non-trivial”σ-ideal of closed setsI⊆U
0 is non-Borel. This implies both the result of Solovay, Kaufman that bothU
0 andU (theσ-ideal of closed sets of uniqueness) are not Borel as well as the theorem of Debs-Saint Raymond that every Borel subset ofT of extended uniqueness is of the first category. A further extension to ideals contained inU
0 is given.