Differentiable Surfaces and Boundaries of Regions in Differential Geometry
Region1
Consider a subset $\mathscr{R}$ of a surface $M$. If $\mathscr{R}$ is an open set, and for any two points in $\mathscr{R}$ there exists a curve on $\mathscr{R}$ containing both, then $\mathscr{R}$ is called a region of $M$.
Boundary
For a region $\mathscr{R}$ of a surface $M$, the following set $\partial \mathscr{R}$ is called the boundary of $\mathscr{R}$.
$$ \partial \mathscr{R} = \left\{ p \notin \mathscr{R} : \exists \left\{ p_{j} \right\} \subset \mathscr{R} \text{ such that } \lim\limits_{j \to \infty} p_{j} = p \right\} $$
Curve Enclosing a Region
If the image of curve $\boldsymbol{\gamma}$ is the boundary of region $\mathscr{R}$, and the intrinsic normal $\mathbf{S}$ of $\boldsymbol{\gamma}$ points inside while $-\mathbf{S}$ points outside $\mathscr{R}$, then it is said that curve $\boldsymbol{\gamma}$ bounds a region $\mathscr{R}$.
Explanation
The intrinsic normal $\mathbf{S}$ pointing inside means that the curve must rotate counterclockwise. For example, for a surface $M = \mathbb{R}^{2}$, $\mathscr{R} = \left\{ (x,y) \in \mathbb{R}^{2} : x^{2} + y^{2} \lt 1 \right\}$ is a region of $M$. Also, curve $\boldsymbol{\alpha}(\theta) = (\cos \theta, \sin \theta)$ is the boundary of $\mathscr{R}$.
On the other hand, let’s say the surface $M$ is a torus $T^{2}$ as in the right picture below. If all points except the image of $\boldsymbol{\gamma}$ are considered the region $\mathscr{R}$, then the boundary of $\mathscr{R}$ becomes the image of $\boldsymbol{\gamma}$. However, since $-\mathbf{S}$ of curve $\boldsymbol{\gamma}$ does not point outside but inside $\mathscr{R}$, it is said that curve $\boldsymbol{\gamma}$ does not bound $\mathscr{R}$.
Richard S. Millman and George D. Parker, Elements of Differential Geometry (1977), p180-181 ↩︎