Punctured Riemann surface

Removing a closed discrete set from a Riemann surface leaves an open complex one-manifold; if connected, it is again a Riemann surface.

A path in a surface can be perturbed inside coordinate discs to avoid finitely many prescribed points.

The punctured plane ${\mathbb{C}}^{\ast }$ is a connected Riemann surface homeomorphic to a cylinder.

Polar coordinates give ${\mathbb{C}}^{\ast }\cong S^1\times \mathbb{R}$ after taking logarithmic radius.