In mathematics, Grothendieck's connectedness theorem states that
if A is a complete local ring whose spectrum is k-connected and f is in the maximal ideal, then Spec(A/fA) is (k − 1)-connected. Here a Noetherian scheme is called k-connected if its dimension is greater than k
and the complement of every closed subset of dimension less than k is connected. Grothendieck XIII.2.1

It is a local analogue of Bertini's theorem.