Open and Closed Sets in Euclidean Space

# Open and Closed Sets in Euclidean Space

Recall from the Interior, Boundary, and Exterior Points in Euclidean Space that if $S \subseteq \mathbb{R}^n$ then a point $\mathbf{a} \in \mathbb{R}^n$ is said to be an interior point of $S$ if there exists an $r > 0$ such that $B(\mathbf{a}, r) \subseteq S$. Furthermore, $\mathbf{a}$ is said to be a boundary point of $S$ if for every $r > 0$ we have that there exists $\mathbf{x}, \mathbf{y} \in B(\mathbf{a}, r)$ such that $\mathbf{x} \in S$ and $\mathbf{y} \in S^c$. We also said that a point $\mathbf{a}$ is called an exterior point of $S$ if $\mathbf{a} \in S^c \setminus \mathrm{bdry} (S)$.

We are now ready to formally define the concept of an open and closed set.

 Definition: A set $S \subseteq \mathbb{R}^n$ is said to be Open if $S$ is precisely the set of all of its interior points, i.e., $S = \mathrm{int} (S)$.

For example, if $S$ is defined to be the interior of a half cone intersected by any plane that does not pass through the vertex of the cone, then $S$ is an open set. Of course, sometimes we may want to have a set of points in $\mathbb{R}^n$ that contains both its interior points and its boundary points which we define below.

 Definition: A set $S \subseteq \mathbb{R}^n$ is said to be Closed if $S^c$ is open.

In our example above if $S$ is defined to be the interior and surface of a half cone intersected by any plane that does not pass through the vertex of the cone, then $S$ is a closed set. 