TITLE: How to define "being inside of something" in the context of topology?
QUESTION [13 upvotes]: 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 :).

REPLY [3 votes]: 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.