Assume I have an open, nonempty Set $X \subseteq \mathbb{R}^n$ and a compact set $K \subseteq X$. Intuitively, there must be a simplex $S$ such that $K \subseteq S \subseteq X$. I would love to see a rigorous proof for this.
I am sure there is an open cover of $K$, such that a finite subcover contains a simplex...