-1

I have read several texts that give a very technical description of what a topological space is, but I can't find any notes that really give an intuitive description of what it is and I'm really struggling to get my head around it. Is the idea behind a topology the notion that we wish to be able to have a notion of "nearness" of elements to one another in a set, without introducing a notion of "distance"? By "nearness" is it meant that two elements are within the same neighbourhood (i.e. they lie within the same open subset)? Given this, is the idea that two objects are topologically equivalent, e.g. a coffee cup and a doughnut, that the neighbourhoods that elements "reside" in are unaffected under a continuous deformation from one object to the other, such that "nearby" elements remain "nearby" (relative to one another)?

I would really appreciate it if someone is able to give me an intuitive notion of a topological space, and also a motivation for such a concept. Thanks.

Perpetual learner
  • 599
  • 2
    Possible duplicate: http://math.stackexchange.com/questions/281185/intuition-behind-topological-spaces. See also http://math.stackexchange.com/questions/31859/what-concept-does-an-open-set-axiomatise. – Pedro M. Mar 02 '15 at 14:10
  • Thanks for the links, but I was looking for a very conceptual, intuitive motivation and possible description of a topological space if possible? – Perpetual learner Mar 02 '15 at 14:54
  • Yes, "nearness without distance" is a good first-approximation for how to think about topological spaces. – Jack M Mar 02 '15 at 15:24

1 Answers1

0

This answer may be a bit too philosophical but I hope nobody will be offended. The intuition behing topology is that it generalizes the notion of "shape" in a very rough meaning. Consider the following chain of structures:

  1. Geometry (whatever it is)
  2. Differential structures
  3. Topological spaces
  4. Sets

(It could be longer and finer and branch, but let's keep it simple.) When going down in this list, you see "less and less" details. When studying geometry, you have a world with lines, angles, distances, triangles.. If you forget how to measure angles and distances (you loose your sextant) you still may see what are tangent vectors (that is, in which direction you can move) or which mountains are just "smooth hills" and which have "sharp peaks".

If you further forget the differential structure, you can no longer use derivatives of any kind, but you still can tell whether you live on a sphere (going in one direction, you eventually come back) or a torus or a non-compact plane. You still can do a "triangulation" (consisting not of straight line segments because you don't know what is a straight line, but of some curves..) and compute the Euler characteristic of your space. This is the "topology".

Further, if you choose to forget even this structure and don't remember what is continuous and what not, the only thing that remains is a set. Then you can no longer distinguish $\mathbb{R}$ and $\mathbb{R}^2$ but you still see a difference between countable and uncountable sets, for example.

So this is one way how to look on topology -- as sitting in the middle between "sets" and some "more geometrical" structures.

Community
  • 1
Peter Franek
  • 11,522