  Consider a $n \times n$ matrix from the Gaussian Unitary Ensemble (GUE).
Given a finite collection of bounded disjoint real Borel sets $(\Delta_{i,n},\
1\leq i\leq p)$, properly rescaled, and eventually included in any
neighbourhood of the support of Wigner's semi-circle law, we prove that the
related counting measures $({\mathcal N}_n(\Delta_{i,n}), 1\leq i\leq p)$,
where ${\mathcal N}_n(\Delta)$ represents the number of eigenvalues within
$\Delta$, are asymptotically independent as the size $n$ goes to infinity, $p$
being fixed.
  As a consequence, we prove that the largest and smallest eigenvalues,
properly centered and rescaled, are asymptotically independent; we finally
describe the fluctuations of the condition number of a matrix from the GUE.
