Draw two balls centered at the origin in R^3, where $r_1 < r_2$. What is the topological name for the boundary created? Does it even make sense to talk about a figure with an inside boundary and an outside one?
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Please let me know if the following answer is at the correct level. Thanks!
If you have a "nice" subset $X \subset R^3$ then it makes sense to talk about the outer boundary of $X$ and the inner boundaries. Here "nice" means a connected, compact submanifold of dimension three.
Let's break this down into easy pieces.
If $F \subset R^3$ is a compact, connected surface without boundary then $F$ cuts $R^3$ into exactly two pieces $A$ and $B$ so that the closure of $A$ is compact and the closure of $B$ is not compact. Let's refer to $A$ as the piece "inside" of $F$ and to $B$ as the piece "outside" of $F$. (For help with the exercise please see Hatcher's notes on three-manifolds.)
Useful Lemma: If $F$ and $F'$ are surfaces (as above), and $F$ meets the outside of $F'$, then $F'$ meets the inside of $F$. Proof: Exercise.
Now suppose that $X$ is as above. Let $F_0, F_1, \ldots, F_k$ be the connected components of $\partial X$, the boundary of $X$. Let $A_i$ and $B_i$ be the inner and outer pieces of $R^3 - F_i$. Say that $F_i$ is "inside" $X$ if $X$ meets $B_i$ (and so is disjoint from $A_i$). Otherwise say that $F_i$ is "outside" of $X$. (Exercise: check everything is well-defined using connectedness of $X$!)
Theorem: exactly one of the $F_i$ is outside $X$ and the rest are inside. Proof: Exercise.
Ok, this is a long row to hoe. But this gives precise meanings to inside and outside and then verifies your intuition about those words. It is also comforting that it all comes down to understanding just a few facts -- the Schoneflies theorem (Alexander's theorem in dimension three) and properties of connected sets.
Does a plane have two sides? Can the hole be enlarged and stretched so that the inner side of the sphere is everted? The resultant structure should be equivalent to a disc, with Euler characteristic of one. However, the surface is probably polarized, with opposite curl values on either side. this would reflect the opposite vector fields on the spherical surfaces, when viewed from the perspective of an outside or inside observer, relative to the sphere. so without vector fields you don't know which end is up. This is a state characteristic of human beings at their present stage of development.