# How to define “being inside of something” in the context of topology?

I'm a Psychologist and Neuroscientist with interest in math and I just started reading about Topology. I have to say it's not easy to grasp the concepts without a practical example, so I'm trying to understand topology in a practical (psychologically applicable) way.

I was thinking for example about the concept of something being inside of another thing, like someone being inside a house, tea being inside a cup or a smaller circle lying inside a bigger one asf. Humans can identify those things as being the same (belonging to one equivalence class?), i.e. if I ask someone to identify the object inside the other one, every normal functioning person will be able to identify the object inside, no matter how different the properties (color, size, form asf.) of the objects are. So there must be some general properties the brain uses.

But how can I define this concept of being inside another thing topologically/mathematically so that it is applicable for a wide range of objects?

And what if it gets even more complex. What if a time factor is included like putting something inside another thing. For example putting a key inside a keyhole, putting a steak in the frying pan, putting food into a shopping bag asf. So here it's about a processes over time which should belong to the same equivalence class.

How can this be defined?

I hope it became clear what I mean and I'm looking for some inspirational thoughts. Also if anyone can recommend literature with emphasis on practical applications, I'd be thankful :).

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Why do you expect these concepts to belong to topology? The first ("being inside") is simply what sets are all about. – Tobias Kildetoft Feb 5 at 13:29
Tea isn't inside a cup according to topology... – Jp McCarthy Feb 5 at 13:29
The kinds of examples of being inside you refer to are not really places for topology to shine. Homeomorphism, the concept of "these are the same as far as topology is concerned" does not preserve things like "food being in a shopping bag" (though arguably "food touching a shopping bag"). If you're looking for a mathematical idea that's not necessarily from Topology, perhaps "inside" is close to "is a subset of the convex hull", but that's a geometric concept, not a topological one. – Mark S. Feb 5 at 13:40
@amcalde ...but that is not true under homeomorphism and so not a topological property of a cup. – Jp McCarthy Feb 5 at 14:04
@JpMcCarthy This is why I am more of a computational geometer than a topologist! – amcalde Feb 5 at 14:26

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 $X$.

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Thanks Lee, I have some intuition about that now, although it's not that easy to understand for me so far :). But that's ok, i just wanted to have some insight into higher mathematics and how "it is done" by mathematicians. Now I can start to read upon some of the ideas behind it. One thing that I miss though, is the relation to time, like with my "putting something inside another thing" example. Can you give me a hint regrading this and what to read upon or where to look in order to gain some more insight? – holistic Feb 5 at 15:14
I'm not sure what to suggest regarding your examples involving time. At a particular frozen moment of time, your key example and your shopping bag example could be regarded as examples consistent with the "convex hull" notion. You could then consider that the concept of "inside" could vary as time advances: for these intervals of time the key is not inside the lock; for those intervals of time the key is inside the lock. – Lee Mosher Feb 5 at 15:34
But here's a warning. I don't think you'll get any one consistent notion of "inside" to fit all of the examples in your question. If you clearly understand the three mathematically formal notions I've listed, plus some notion that incorporates time, it will still probably be easy for you to dream up further examples of "inside" that fall outside of those mathematically formal notions. It's hard to make mathematics that incorporates all of reality! – Lee Mosher Feb 5 at 15:36
As for readings, you might just pick up a first book on topology such as the title "Topology" by Munkres. He covers the Jordan curve theorem, for example (which is the part preceding "Furthermore" in my first bullet). – Lee Mosher Feb 5 at 15:39
Thank you Lee! That's very helpful thinking about it as a change of the inside concept over time, this gave me some insight :). I understand that it will not be possible include all of my examples, I was just looking for some insight into this matter. I downloaded the book by Munkres, but I'm really having trouble to memorize and understand all of those definitions without a practical example. I also found the book Topology and Its Applications by Basener might seems promising as well. Thanks again! – holistic Feb 5 at 15:43

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.

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Thanks for the answer Hugh. I'm especially interested to gain some insight into how mathematicians think about time geometrically like with my "putting something inside another thing" example. Although I probably won't understand it yet, some idea where to look might be helpful for me to read upon the subject – holistic Feb 5 at 15:11

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.

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Thank you Emilio! I like your answer very much, since it now gives me some insight into what part of mathematics/topology might be suited to answer such questions, especially the notion of something moving in time. Can you maybe answer what type of manifold a mathematician would use to describe "putting a key in a keyhole" (or another easier example) in an abstract way? Do they have to be invented for a specific purpose or are there some heavily used manifolds which can be used for most purposes? – holistic Feb 5 at 20:38
I suppose that a door with a keyhole can be thinked as topologically equivalent to a torus in $3D$, where the keyhole is the hole of the donut. In this case a line that describe a key that enters the keyhole is part of a loop that is topologically characterized by the fact that it is not reducible to a point without cut the surface of the torus. But note that it's only a suggestion :) – Emilio Novati Feb 6 at 14:00
Interesting, thank you :)! – holistic Feb 6 at 14:58

The notion of subset of a set may suffice in many cases, but I think your question is more related to the definition of the inside region defined by a Jordan curve, which relies on a difficult result (Jordan curve theorem).

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