Special points

Figure 4.1: An illustration of the boundary definition. Suppose $X = {\mathbb{R}}^2$, and $U$ is a subset as shown. Three kinds of points appear: 1) $x_1$ is a boundary point, 2) $x_2$ is an interior point, and 3) $x_3$ is an exterior point. Both $x_1$ and $x_2$ are limit points of $U$.

From the definitions and examples so far, it should seem that points on the ``edge'' or ``border'' of a set are important. There are several terms that capture where points are relative to the border. Let $X$ be a topological space, and let $U$ be any subset of $X$. Furthermore, let $x$ be any point in $X$. The following terms capture the position of point $x$ relative to $U$ (see Figure 4.1):

• If there exists an open set $O_1$ such that $x \in O_1$ and $O_1 \subseteq U$, then $x$ is called an interior point of $U$. The set of all interior points in $U$ is called the interior of $U$ and is denoted by $\operatorname{int}(U)$.
• If there exists an open set $O_2$ such that $x \in O_2$ and $O_2 \subseteq X \setminus U$, then $x$ is called an exterior point with respect to $U$.
• If $x$ is neither an interior point nor an exterior point, then it is called a boundary point of $U$. The set of all boundary points in $X$ is called the boundary of $U$ and is denoted by $\partial U$.
• All points in $x \in X$ must be one of the three above; however, another term is often used, even though it is redundant given the other three. If $x$ is either an interior point or a boundary point, then it is called a limit point (or accumulation point) of $U$. The set of all limit points of $U$ is a closed set called the closure of $U$, and it is denoted by $\operatorname{cl}(U)$. Note that $\operatorname{cl}(U) = \operatorname{int}(U) \cup \partial U$.

For the case of $X = {\mathbb{R}}$, the boundary points are the endpoints of intervals. For example, 0 and $1$ are boundary points of intervals, $(0,1)$, $[0,1]$, $[0,1)$, and $(0,1]$. Thus, $U$ may or may not include its boundary points. All of the points in $(0,1)$ are interior points, and all of the points in $[0,1]$ are limit points. The motivation of the name ``limit point'' comes from the fact that such a point might be the limit of an infinite sequence of points in $U$. For example, 0 is the limit point of the sequence generated by $1/2^i$ for each $i \in {\mathbb{N}}$, the natural numbers.

There are several convenient consequences of the definitions. A closed set $C$ contains the limit point of any sequence that is a subset of $C$. This implies that it contains all of its boundary points. The closure, $\operatorname{cl}$, always results in a closed set because it adds all of the boundary points to the set. On the other hand, an open set contains none of its boundary points. These interpretations will come in handy when considering obstacles in the configuration space for motion planning.