How to define "being inside of something" in the context of topology?

Mathematics expropriates many terms of ordinary language. Different branches of mathematics expropriate the same term in different ways. And where confusion really arises is where one branch of mathematics --- say, topology --- depends on another branch of mathematics --- say, set theory --- but those two branches use the term in different ways.

Your word "inside" is like that.

The set theoretic relation $A \subset X$ can be read with high formality as "$A$ is a subset of $X$", or with low informality as "$A$ is inside $X$".

The Jordan/Schonflies theorem is a result in topology which uses the word "inside" in a different manner, but which also uses lots of set theoretic terminology, inviting lots of confusion if one wanders outside of the more highly formal language. Here is what that theorem says in high formality:

• If $c \subset \mathbb{R}^2$ is homeomorphic to the circle $S^1$ then $\mathbb{R}^2-c$ has two components, called the inside $C_{in}$ and the outside $C_{out}$, which are distinguished from each other by the property that the closure $\overline C_{in}$ is compact whereas the closure $\overline C_{out}$ is noncompact. Furthermore, there is a homeomorphism $f : \mathbb{R}^2 \to \mathbb{R}^2$ such that $f(c)$ equals the unit circle $S^1$, $f(C_{in})$ is the open unit ball consisting of points at distance $<1$ from the origin, and $f(C_{out})$ is the subset of points at distance $>1$ from the origin.

And, here is what the Jordan/Schoenflies theorem says in low formality (and with loss of some information):

• A circle in the plane has two complementary components, an inside and an outside. The inside is an open ball, whose closure is a closed ball having the original circle as its boundary.

So then, having this theorem in my hand, I can formulate statements like your example of "this circle is inside that circle", remembering that to make concrete mathematical sense of the sentence I can revert to the high formality version of the Jordan/Schoenflies theorem.

Finally, as suggested in the comment of @MarkS, there is a third and still different concept of "inside" which fits some of the examples of your question, and which is formulated by making use of the first set-theoretic notion of "inside", namely the subset concept: Given subsets $A,X \subset \mathbb{R}^n$, we can say that $A$ is inside $X$ if $A$ is a subset of the convex hull of $X$.

Here is a simple answer (perhaps too simple). Inside and outside are partitions of a space of some number of dimensions. The informal idea is that any two points "inside" a shape can be joined by a continuous (not necessarily straight line) that does not intersect with the shape's boundary. A point is said to be "outside" the shape if it cannot be connected to a point that is "inside" the shape without crossing the boundary of the shape. There are lots of other considerations here that I am leaving out, but this is the very basic idea.

You have actually used the word "inside" in several different mathematical senses. Imagine a two dimensional circle floating in three dimensional space with a line passing through it. We might informally say the line is "in" the circle but the term is just that: informal. It is more correct to say that tea is "on" a teacup because it is not actually enclosed and only the "accident" of gravity is keeping it there.

People tend to use "inside" to mean something like: "there exists a two dimensional plane in which the cross-section of object a is inside the cross-section of object b". For example, if you consider the door as a two dimensional plane then the cross section of the key would be inside the area defined by the keyhole. I hope that's clear.

Time is another matter altogether. There are ways of thinking about time geometrically but why not stay with two or three dimensions until you feel you have mastered that.

As noted in other answers, mathematics use different notions of ''inside/outside'' or ''interior/exterior''. And probably none of them completely capture the meaning of the usual language. So, instead of starting from mathematical definitions, I try starting from the intuitive meaning of '' inside/outside''. It seems to me that the idea of being inside or outside something require at least two conditions:

1) that such thing is inserted on some greater ''ambient'' so that there can be an ''outside'' .

2)That it has a ''boudary''

I give some example: A circle ( the boundary) in a plane (the ambient) divides the plane in two non connected components and we can define the interior as the component that contains the center of the circle and the exterior as the other component. But, what about if the ambient is a sphere (as the Earth)? A circle on a sphere can hawe two ''interiors'' that can be difficult to distinguish: thik at the equator as a circle, what is its interior? So it seems that the common intuition of ''interior/exterior'' assumes (unconsciously?) that the ambient is isomorphic to a $\mathbb{R}^3$ space.

But the example of the cup of tea suggest that this intuitive ambient space is really a physical space that has a privileged direction up-down so that the tea is in the cup if it is concave up, but it comes out if we reverse the cup.

Now, how we can define such intuitions in mathematical way? I think that we can find the mathematical concepts that can work better in the theory of topological manifolds. Here the concepts of connected components, boundary, embedding in a greater space, ... can be well defined (also if not always in a simple way).

If we want to describe the motion of something from outside to inside a set delimited by a boundary, we have to use some function of time, so we need some property of continuity and differentiability for such a function and, probably, we have to work in a differentiable manifold, so that we can find if a line that represents the motion intersect the boundary and in wich direction.

Finally, I really don't know how to treat the existence of a privileged direction, but someone more expert in topology probably knows how to do.