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\n\"Rewriting\" an (oriented) surface
\nLet us first state a classical theorem. It can be found in Topology from an Advanced\nGeometric Viewpoint by R.M.Smale\nLemma\nLet $S$ be an oriented surface. Then we can find an oriented surface $N$, such that:
\nThere is an orientation preserving homeomorphism between $S$ and $N$,\nThere is an orientation preserving diffeomorphism between $S$ and $N$.
\nIn fact, we can assume that $N$ is a cube and it suffices to find, for each face, an orientation preserving homeomorphism to cover the corresponding face of $S$.\nProblem\nLet us continue. Let $S$ be a closed connected oriented surface. Fix an orientation on $S$.\nQuestion\nIs it possible to \"rewrite\" $S$ (without changing its orientation) so that it becomes equal to a plane in its re-written version?\nBasically, I want to find a sequence of homeomorphisms that shrink the surface $S$ while preserving the orientation. I also want the homeomorphisms to \"really\" shrink the surface, so I need continuity.\nP.S. I apologize for this \"general\" question. If it seems it is too general, then I will add more details in my comment at your answer.
\nA:
\nAnother possible answer is through the use of self-indexing triangulations. A self-indexing triangulation of a closed 2-manifold is basically just a collection of smaller versions of the surface. As such, you can'swap out' a triangle for a square to basically get a rectangle that's already a plane. However, a self-indexing triangulation of an open surface must be a regular one, which is also a plane. A regular self-indexing triangulation of a surface is a subdivision of the surface into triangles, and is called a mesh.
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