How to denote the inside of a manifold?

Question

In $\mathbb{R}^3$, I want to denote the inside of a closed surface $S$. Now I could define a volume $V$ such that $A = \partial V$, but I do not want to introduce an unnecessary, additional symbol $V$ and instead use something like $\text{inside}(A)$. For example, when using the Gauss integration theorem

$$\oint_A \boldsymbol X \cdot d\boldsymbol S = \int_{\text{inside}(A)} \nabla\cdot\boldsymbol X \ dV.$$

I saw a similar notation in Richard Feynman's lectures on physics. Is there no elegant, topological way to denote the inside?

Answer 1

It's a tricky point to get across, primarily because there is a deep mathematical theorem in the background, the Jordan Brouwer separation theorem. Although that wikipedia link uses the words "interior" and "exterior", those words have other meanings which causes ambiguity, as some of the comments point out. A better terminology would be one that is just as descriptive but avoids ambiguity, and in that regard I like "inside" and "outside".

But to answer your actual question, no, there is no standard, elegant, topological notation or terminology. I actually quite like the way you wrote it in your question with "inside(A)".

Answer 2

Please take it as just an observation.

I think that one can denote the interior of a closed surface exploiting the fact that $ S $ closed surface divides the space (which in this particular situation is the Euclidean 3D space) in 2 open sets and assuming to use polar coordinates to identify each point, the interior is the subspace not containing the $ r = \infty $ elements