dg.differential geometry – Is it a nil hyper surface locally achronal?

Let (M, g) be a Lorentzian variety. A subset A of M is acronym if there is no time curve joining two points in A.
A subset A is locally local if, for any point in A, there is an open neighborhood U of p such that $ A cap U $ It is achronal.

My question is: if L is an embedded null surface, can we make sure it is locally local? In the affirmative case, is there any reference to this?

Thank you.