Topology of shapes I am beginner in this subject. I understand at least definition of topology. However, I don't get how Topology: a Set with topology on that Set accounts for shapes. How would you describe the topologically a torus? or Sphere? Is it possible to represent it in the form $(X,S)$, what will $X$ and $S$ be for a torus?
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 A: A topology is something quite abstract. For many geometric shapes, you can start with something more intuitive: Distance
Whenever you have a notion of measuring distance on a shape, you can turn this into a topology. Let's first look at what to call a distance: Denote the distance between two points as $d(x,y)$, then we require some intuitive constraints


*

*Distance is never negative and $d(x,y) = 0 \Leftrightarrow x=y$.

*Distance is symmetric, i.e. $d(x,y) = d(y,x)$.

*There are no "shortcuts", i.e. $d(x,y) + d(y,z) \geq d(x,z)$.


A set with such distance function $d$ is called a metric space. So by just considering shortest pathes on a torus, a sphere ..., you have made them into metric spaces. 
Now what's our topology? Having distance defined, we can look at the set of points closer than some radius $\epsilon$ and we call $B_\epsilon(x) = \{y : d(x,y) < \epsilon\}$ a ball around $x$ with radius $\epsilon$. By definition, we now call a subset $A$ of our metric space open iff we can find a tiny $\epsilon$-ball around every point in it, so our topology is
$$ \tau = \{ A \subset X | \forall x \in A \exists \epsilon > 0: B_\epsilon(x) \subset A \} $$
You can check that this defines a topology on your metric space.
A: One standard approach to the torus is as follows. 
First, define a surface to be a compact topological space where every point has a neighborhood homeomorphic to an open disc. 
Then, prove that every surface is homeomorphic to a polygon with edges identified in pairs. 
Then define a torus to be a surface homeomorphic to a square with opposite pairs of edges identified. Well, you have to be a little careful about the orientation of the edges, or else you're liable to get a Klein bottle; the orientations have to be "opposite", in the sense that if you travel one edge of a pair in the chosen orientation, and continue past the edge, around the square, when you reach the other edge in the pair you are going against the orientation. 
For a sphere, you can take a "polygon" with two edges (this is topology, not geometry, so such a thing makes sense), and identify the two edges, oppositely oriented.  
