Let $\Omega\subset\mathbb{R}^N$ be an open, bounded and connected Lipschitz domain. Is it true that we can find some $R>0$ such that any $N$-dimensional open ball $B(x,r)$ with $r\leq R$ that intersects $\Omega$ has the property that $\Omega\cap B(x,r)$ is a open, bounded and connected Lipschitz domain too?


I don't know any reference that specify those facts and I decided to ask here.


[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/GPf5V6sQ.png

**EDIT:** *I look at the solutions and see that the problem is that we lose **connectedness**.*

**Question: If the intersection between an open ball and a Lipschitz domain is connected, then we can say that the intersection is a Lipschitz domain?**