I have defined sides of a surface (represented by such things as a set in a topological space) purely topologically. I also gave two (possible non-equivalent) definitions of special points of a surface (such “singularities” as points of the border of a closed disk).
Currently these definitions and questions are presented in the file addons.pdf.
It is very interesting and intriguing. The theory is considered not in the framework of topological spaces, but in more general theory of funcoids.