We often define torus or protective map by some standard gluing of edges (quotient map) of polygon. But how to prove that the image of this quotient map is indeed a 2-manifold (Hausdorff, second countable and locally homeomorphic to unit disc)?
To ask it in a clearer way, assume $P$ is a polygon with even number of edges, and the edges are pairwise glued together by the quotient map. Then how to prove the image is a surface?
At points which do not lie on edge of the polygon the conditions become clear once you restrict to small basis elements which avoid the edge, so it only remains to deal with the edges of the polygon. The edges are actually not too bad either. On the square representing the torus, neighborhoods of points on the edge are two half disks with boundary on either side of the square. For any two points on the edge, we can choose little pairs of half disks to show that the edge is Hausdorff. In addition, points on not the edge can also be separated from points on the edge by a little disk and little half disks. Pairs of half disks with rational center (viewing each edge as [0,1]) and rational radius show that the space is second countable. Finally, two half disks glued together make a whole disk which gives our local homeomorphism condition. All of this generalizes with no more work to arbitrary even edged polygons.
