  In this paper we study the $\nu$-continuity of the spectrum and some of its
parts. We show that the approximate point spectrum $\sigma_{ap}$ is upper
semi-$\nu$-continuous at every Fredholm operator, then we give sufficient
conditions to guarantee the $\nu$-continuity of $\sigma_{ap}$. Also we show
that the restriction of the Weyl spectrum on the class of essentially $G_1$
operators is $\nu$-continuous. Finally, we investigate the $\nu$-continuity of
the spectrum on the class of $p$-hyponormal operators.
