Closed sets

After reading this lesson, you will be able to define and give examples of closed sets. Intuitively, a closed set is a set which has some boundary. A more formal definition is below.


Definition and examples


Definition: We say that a set $B \subseteq \mathbf{R^{n}}$ is a closed set if its complement $\mathbf{R^{n}} \setminus B$ is an open set.

Recall that we defined open sets and gave their examples in the lesson Open sets. Therefore, we can immediately give a few examples of closed sets.

Example 1: A set

$$\mathbf{R^{n}} \setminus \{y \in \mathbf{R^{n}} : d(x, y) < r\}$$

is a closed set because an open ball $K(x, r) = \{y \in \mathbf{R^{n}} : d(x, y) < r\}$ is an open set.


Example 2: Point in $\mathbf{R^{n}}$, i.e. singleton $\{x\}, x \in \mathbf{R^{n}}$ is closed in $\mathbf{R^{n}}$.


Example 3: A set $\overline{K}(x, r) = \{y \in \mathbf{R^{n}} : d(x, y) \leq r \}$, where $x \in \mathbf{R^{n}}, \ r > 0$ is a closed set. We call $\overline{K}(x, r)$ a closed ball with center x and radius r.


The difference between an open ball and closed ball in $\mathbf{R^{n}}$ is that closed ball also contains all the ”exterior” points. For instance, a closed ball in $\mathbf{R^{3}}$ contains all the points on the surface of and in the sphere.


Example 4: Some sets are neither open nor closed. For instance, a half – open interval $\left<0, 1\right]$.


Example 5:  In the lesson Open sets we mentioned that sets $\emptyset$ and $\mathbf{R^{n}}$ are open. Furthermore, both of those two sets are closed also. Moreover, those are the only two sets in $\mathbf{R^{n}}$ that are open and closed at the same time.

Example 6: Every segment $[a, b]$ is closed in $\mathbf{R}$.


The closure of a set


Definition: Let $A \subseteq \mathbf{R^{n}}$. A closure of a set A, denoted by $\overline{A}$ or $\mathcal{Cl}A$, is an intersection of all closed sets which contain set A. In other words,

$$\overline{A} = \underset{\underset{F \ closed}{F \supseteq A}}{\bigcap} F.$$

Note: As the intersection of closed sets is a closed set, we conclude that $\overline{A}$ is a closed set. Also, it is clear from the definition that $A \subseteq \overline{A}$. Furthermore, a closure of a set A is the smallest closed set which contains set A. Moreover, a set A is closed if and only if $A = \overline{A}$.

Example 7:

a) $\overline{\left<0, 1\right>} = [0, 1]$

b) $\overline{\left[0, 1\right> \cup \{2\}} = [0, 1] \cup \{2\}$.

Note: A closure of a set A is the union of set A and its derivative set. In other words, $$\overline{A} = A \cup A’.$$


The boundary of a set


Definition: Let $A \subseteq \mathbf{R^{n}}$. A boundary of a set A is given with

$$\partial A = \overline{A} \cap \overline{\mathbf{R^{n}\setminus A}}.$$

Note: As we can see, a boundary of a set is an intersection of closed sets. Therefore, a boundary of a set is a closed set also. Furthermore, notice that $\partial A = \partial (\mathbf{R^{n}} \setminus A)$.


Example 8: $\partial K(x, r) = S (x, r) = \{y \in \mathbf{R^{n}} : d(x, y) = r\}$ ( a sphere with with center x and radius r)