How do you render the union of a simple closed curve and its interior?

Question

In this question there is confusion about the answer because it appears (to me, anyway) that the origin was supposed to be rendered as not on or inside a simple closed curve; but the question is being answered as if the origin were merely not on the curve. This causes the problem to be ill-posed. How should the origin have been rendered precisely as not inside or on the curve?

Answer 1

With the notation set up in that problem, I don't know of a way in notation alone.

In words, it's easy enough to say let $\gamma :[a,b]\to\Bbb{C}$ be a simple closed curve with $0$ not on $\gamma$ or in its interior.

Note

As pointed out by Ted Shifrin in the comments, in my original answer, I answered your question, which assumes that the curve is simple closed, but the linked question only assumes closed, and I ended up writing merely closed in my answer originally. That the curve is simple closed is important for the existence of the interior.