Intersection of $max(R)$ with a closed subset in $Spec(R)$

Question

Let $R$ be a commutative ring with unity and $E$ be a nonvoid closed subset of $Spec(R)$. If $U$ is an open subset of $Spec(R)$ with $E∩Max(R)⊆U$, where $Max(R)$ is the set of maximal ideals of $R$, is it necessarily true that $E⊆U$? I think that we must use the topology induced by the varieties $V(A)$ when $A$ are the ideals of $R$. Thanks in advance for any help.

Answer 1

I think this is true. If there were and ideal $\mathfrak{a} \in E\cap U^c$ then there would be a maximal ideal $\mathfrak{m} \supseteq \mathfrak{a}$. Since both $E$ and $U^c$ are closed, $\mathfrak{m}\in E\cap U^c$ contrary to assumption.