📂Geometry

Simple Connected Region

Definitions

Let $\mathscr{R}$ be a region of the surface $M$. If every closed curve within $\mathscr{R}$ is null-homotopic, then $\mathscr{R}$ is said to be simply connected.

Description

Easy examples such as $\mathbb{R}^{2}$, disk $\left\{ x^{2} + y^{2} = r^{2} \right\}$, and sphere $\mathbb{S}^{2}$ are immediately thought to be simply connected. However, as shown in the figure below, one can see that the torus $T^{2}$ is not simply connected. Unlike $\gamma$, $\alpha$ and $\beta$ cannot be contracted to a single point.