Proving OP(L) is regular

Question

i am currently working on a proof to show that if L is regular then OP(L) is also regular without the use of pumping lemma. For the sake of this question, i am using {0,1} as alphabets.

OP(L) = {w| wz ∉ L for every z ∈ {0,1}* - {∈} }

At the moment, i have a hard time starting the proof. What i understand is that, If L is regular then we can define a DFA that can decide L. And then we can define another machine for OP(L) with every transition from M along with ∈ transitions.

Am i on the right track? I could really use some hints.

Answer 1

Hint. Use the fact that regular languages are closed under complement.