On the fundamental groups of perforated surfaces

A perforated surface is the complement $\mathring{\mathrm{\Sigma }}:=\mathrm{\Sigma }\setminus A$ of a countable dense subset $A$ in a connected paracompact surface $\mathrm{\Sigma }$. It is known that the topological type of $\mathrm{\Sigma }\setminus A$ is independent of the choice of $A$. Any perforated surface is one-dimensional, connected, locally path connected, and is not semi-locally simply connected at any of its points. In this paper we obtain a classification theorem for perforated surfaces, using the classification theorem for surfaces. We show that any connected covering of a perforated surface $\mathring{\mathrm{\Sigma }}$ arises from a covering of a surface ${\mathrm{\Sigma }}^{\mathrm{\prime }}$ such that $\mathring{\mathrm{\Sigma }}\cong {\mathring{\mathrm{\Sigma }}}^{\mathrm{\prime }}$. We show that the fundamental group of perforated surfaces are large. We also show that the fundamental groups of $\mathring{\mathrm{\Sigma }}$, the Sierpiński curve and the Menger curve are not Hopfian.