What is the geometric interpretation of the following set? Let $C$ be a circle (with arbitrary radius). Also there are two points $p_1$ and $p_2$ which can move along this circle freely. Let us choose a parametrization of this circle just by choosing polar angle $\phi$ as parameter. Then position of each points $p_1$ and $p_2$ can be denoted by $\phi_1$ and $\phi_2$ respectively. OK. Let ask the following question:
If we consider all possible positions of the above two points we get a set (which we shall call $T_{1,2}$) of pairs 
$$ T_{1,2} = \{(\phi_1, \phi_2)\,\lvert\,  \phi_1, \phi_2 \in [0,2\pi]\}.$$
What is the geometrical interpretation of $T_{1,2}$?
It is easy because $T_{1,2}$ can be considered as Cartesian product of two circles $C_1$ and $C_2$:
$$T_{1,2} = C_1 \times C_2,$$
which is torus. BUT. What it would be if the points $p_1$ and $p_2$ were indistinguishable in the previous example. What the geometrical interpretation of all possible positions of this points it might be?
 A: 
If we consider all possible positions of the above two points we get a set (which we shall call $T_{1,2}$) of pairs 
  $$ T_{1,2} = \{(\phi_1, \phi_2)\,\lvert\,  \phi_1, \phi_2 \in [0,2\pi]\}.$$
  What is the geometrical interpretation of $T_{1,2}$?
  It is easy because $T_{1,2}$ can be considered as Cartesian product of two circles $C_1$ and $C_2$:
  $$T_{1,2} = C_1 \times C_2,$$
  which is torus.

Strictly speaking, the torus $\Bbb T^2$  is a quotient space of the square $ T_{1,2} = \{(\phi_1, \phi_2)\,\lvert\,  \phi_1, \phi_2 \in [0,2\pi]\},$ where for each $x\in [0,2\pi]$ we glue together the boundary points $(0,x)$ and $(2\pi,x)$ and also the points $(x,0)$ and $(x,2\pi,x)$. So if the points $p_1$ and $p_2$ are indistinguishable, we also have to glue together points $(x,y)$ and $(y,x)$ for each $x,y\in [0,2\pi]$. When we glue them we obtain a triangle. When we next glue its boundary points, we should obtain the Möbius strip. Unfortunately, my geometric imagination is weak and I don’t see this, but there are some mentions about it. For instance, in Wikipedia article about Möbius strip it is said to be the configuration space of two unordered points on a circle. 
