prove path connected crossing line

Question

If $U\subseteq \mathbb{R^n} $ and path connected,

with $V\subseteq_{open} U$ , $U\backslash V\subseteq_{open} U$

then $ V=U $or $\phi $

Here are my thoughts, but not working.

Let $ x \in V$ and $ y \in U\backslash V $, and let a path x to y .

When the path go from x to y, there must have a problem at crossing line.

I try to proof by the thought, but I failed.

Can someone give me details?