Topology

An open \(n\)-ball \(\require{physics} \mathcal{B}(\vb{a}, \vb{r})\) is the set \[\begin{equation} \mathcal{B}(\vb{a}, \vb{r}) = \set{\vb{x} \in \mathbb{R}^n ; \thinspace \thinspace||\vb{x} - \vb{a}|| < \vb{r}} \thinspace . \end{equation}\] A point \(\vb{a}\) is called an interior point of \(S \subset \mathbb{R}^n\) if there exists an open \(n\)-ball such that .

A set \(S \subseteq \mathbb{R}^n\) is called open if every point inside it is an interior point of \(S\). Examples are in 1D an open interval, and in 3D the sphere without boundary.

A neighborhood of a point \(\vb{a}\) is an open set \(S\) in which \(\vb{a}\) lies.

A point \(\vb{x}\) is called an exterior point of \(S \subset \mathbb{R}^n\) if there exists an open \(n\)-ball that does not contain any points of \(S\).

A point \(\vb{b}\) is called a boundary point of \(S\) if it is not interior nor exterior. The set of all boundary points of a set \(S\) is called the boundary and is denoted by \(\partial S\).