Let $M = (Q, \Sigma, \delta ,q0,A)$ be an $\epsilon$-NFA. Suppose that there are sets $S \subseteq Q$ and $T\subseteq Q$ such that $ S=\epsilon (S)$ and $T=\epsilon (T)$.
Prove that $S \cap T = \epsilon (S \cap T)$.
My reasoning:
Since $ S=\epsilon (S)$ and $T=\epsilon (T)$We see that S and T have no epsilon transitions coming out of it's states. If there are elements in $S\cap T$, then the states in this intersection will also have no epsilon transitions available. Therefore $S \cap T = \epsilon (S \cap T)$.
I have no idea if this is correct or how to formalize it.