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.

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.

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?

Let $\Omega\subset\mathbb{R}^N$ be an open and connected Lipschitz domain. Consider an $N$-dimensional open ball $B(x,r)$ that intersects $\Omega$. Is it true that $\Omega\cap B(x,r)$ is a Lipschitz domain too?

It is obvious if $B(x,r)\subset\Omega$, but what happens in the remaining case?

More general question: The intersection of two bounded Lipschitz domains is also Lipschitz?

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

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.