For proving Jordan separation theorem in differential manifold theory, one step involves proving the following: Let $z\in{\mathbb{R}^n}\setminus X$, where $X$ is connected, closed manifold of dimensin $n-1.$ suppose that $x\in{X}$ and $U$ is an arbitrary neighborhood of $x$. Then we have to show that there is a point of $U$ that may be joined to z by a curve not intersecting X.\
If we take $S$ to be the set of all those points of $X$ with this property and prove it open and cloed, then using connectedness of $X$ it will be whole of $X$. I have showed it to be closed. Can anyone help me with the openness part of this.