Let $S$ be a set of open sets. Suppose all open sets in $S$ have a common boundary point. That is, $\exists$ a point $p$ such that $\forall s\in S$, $p$ is a boundary point of $s$. It is known that $S$ is an open cover for $Ω$ \ {$p$}, where $Ω$ is a compact set. Is it true that there is always a finite subcover of $S$ for $Ω$ \ {$p$}?
I has thought about it in $\mathbb{R}^n$ and I think that it should be true. But I failed to find a satisfying answer for this. Is there any way to apply Heine-Borel Property for this? Or could this has any ways to solve?
In $\mathbb R\cup\{\infty\}$, the one-point compactification of $\mathbb R$, consider the complements of the closed sets $C_k=\{\infty\}\cup\{n\ge k\mid n\in\mathbb N\}$ for $k\in\mathbb N$. They all have $\infty$ as a boundary point, and they cover $\mathbb R$, but there’s no finite subcover.