My question arise since a pentagon is homeomorphic to a closed disc. This last one is a surface with boundary.
However, a pentagon has vertices, so it seems isn't a 2-manifold.
If you consider a pentagon without edges, it is a 2-manifold, however isn't compact, so isn't a surface.
So... any polygon is a surface?
Regards
A pentagon with boundary is a topological $2-$manifold with boundary, but it is not a smooth $2-$manifold with boundary. So, this depends what you mean by surface.
A pentagon with boundary is a compact topological surface (where we relax the smoothness condition), however it is not a smooth surface in the typical sense. It is, however, a smooth manifold with corners.