Can someone please remind me how this goes?

Here's the idea of proof I'm trying to recall: let $S$ be a closed surface (connected, compact, without boundary) embedded in $\mathbb{R}^3$. Then one can define the "outward-pointing normal unit vector" to $S$ at any point, and subsequently an orientation of the surface.

One would like to define this vector by saying that it points towards exterior points to $S$. So we need some kind of generalization of the Jordan curve theorem saying that the surface cuts $\mathbb{R}^3$ into two pieces (interior and exterior). What is this theorem exactly?

Also, I apologize if this is silly, but is there an obvious argument that a piece of the surface cuts a small tubular neighborhood of it into interior and exterior points (this seems necessary to define the outward normal vector properly)?

Is there a "cleaner" approach to prove this fact? Thanks in advance.