Let $A \subset \mathbb{R}^n$ be a compact set, $\epsilon \in \mathbb{R}_{>0}$, $O := A + \epsilon \mathbb{B}^\circ$, and define the open set
$ \bar O := O \times \mathbb{R}^m. $
Let $C \subseteq \mathbb{R}^n$ be a closed set such that $C \supset O$.
Consider a closed set $\bar C \subseteq \mathbb{R}^n \times \mathbb{R}^m$ such that its projection to $\mathbb{R}^n$ is $C$, i.e. $C$ is the maximal set such that $\forall x\in C \ $ $\exists y \in \mathbb{R}^m$ such that $(x,y) \in \bar C$.
Question: is the set $S := \bar O \cap \bar C$ open?
Comment: as $O \subset C$ we have $\bar C \nsubseteq \bar O$. Seems that the intersection of an open set, $\bar O$, with a closed one, $\bar C$, which is not a subset, should be open as well.