Open Sets and Closed Sets in Metric Spaces

Open and Closed Balls in Metric Spaces

If $\mathbf {a} =(a_{1},a_{2},...,a_{n})\in \mathbb {R} ^{n}$ then the open ball centered at $\mathbf {a}$ with radius $r>0$ is as the set of points $\mathbf {x} =(x_{1},x_{2},...,x_{n})\in \mathbb {R} ^{n}$ contained in:

${\begin{aligned}\quad B(\mathbf {a} ,r)=\left\{\mathbf {x} \in \mathbb {R} ^{n}:\|\mathbf {x} -\mathbf {a} \|<r\right\}=\left\{\mathbf {x} \in \mathbb {R} ^{n}:{\sqrt {(x_{1}-a_{1})^{2}+(x_{2}-a_{2})^{2}+...+(x_{n}-a_{n})^{2}}}\leq r\right\}\end{aligned}}$

Similarly, the closed ball centered at $\mathbf {a}$ with radius $r\geq 0$ is the set of points $\mathbf {x} \in \mathbb {R} ^{n}$ contained in:

${\begin{aligned}\quad {\bar {B}}(\mathbf {a} ,r)=\left\{\mathbf {x} \in \mathbb {R} ^{n}:\|\mathbf {x} -\mathbf {a} \|\leq r\right\}=\left\{\mathbf {x} \in \mathbb {R} ^{n}:{\sqrt {(x_{1}-a_{1})^{2}+(x_{2}-a_{2})^{2}+...+(x_{n}-a_{n})^{2}}}\leq r\right\}\end{aligned}}$

Now notice that $\|\mathbf {x} -\mathbf {a} \|$ is simply the Euclidean distance function $d:\mathbb {R} ^{n}\to [0,\infty )$ for the metric space $(\mathbb {R} ^{n},d)$. We can extend the concept of open and closed balls to any metric space with its own defined metric as defined below.

Definition: If $(M,d)$ is a metric space, $a\in M$, and $r>0$ then the Open Ball centered at $a$ with radius $r$ is defined to be the set $B(a,r)=\{x\in M:d(x,a)<r\}$.

For example, consider the metric space $(\mathbb {R} ^{2},d)$ where the metric $d:\mathbb {R} ^{2}\times \mathbb {R} ^{2}\to [0,\infty )$ for all $\mathbf {x} =(x_{1},x_{2}),\mathbf {y} =(y_{1},y_{2})\in \mathbb {R} ^{2}$ by:

${\begin{aligned}\quad d(\mathbf {x} ,\mathbf {y} )=\mid x_{1}-y_{1}\mid +\mid x_{2}-y_{2}\mid \end{aligned}}$

Then for each $\mathbf {a} =(a_{1},a_{2})\in \mathbb {R} ^{2}$, we can define the open ball centered at $\mathbf {a}$ and with radius $r>0$ to be the set of all points $\mathbf {x} \in \mathbb {R} ^{2}$ contained in:

${\begin{aligned}\quad B(\mathbf {a} ,r)=\{\mathbf {x} \in \mathbb {R} ^{2}:d(\mathbf {x} ,\mathbf {a} )<r\}=\{\mathbf {x} \in \mathbb {R} ^{2}:\mid x_{1}-a_{1}\mid +\mid x_{2}-a_{2}\mid <r\}\end{aligned}}$

We can likewise define the closed ball centered at $\mathbf {a} \in M$ for some metric space $(M,d)$ as follows.

Definition: If $(M,d)$ is a metric space, $a\in M$, and $r>0$ then the Closed Ball centered at $a$ with radius $r$ is defined to be the set ${\bar {B}}(a,r)=\{x\in M:d(x,a)\leq r\}$.

Note the subtle but important difference between the definitions of open and closed balls centered at $a$ with radius $r$.

From the example above, we can define the closed ball centered at $\mathbf {a}$ and with radius $r>0$ to be the set of all points $\mathbf {x} \in \mathbb {R} ^{2}$ contained in:

${\begin{aligned}\quad {\bar {B}}(\mathbf {a} ,r)=\{\mathbf {x} \in \mathbb {R} ^{2}:d(\mathbf {x} ,\mathbf {a} )\leq r\}=\{\mathbf {x} \in \mathbb {R} ^{2}:\mid x_{1}-a_{1}\mid +\mid x_{2}-a_{2}\mid \leq r\}\end{aligned}}$

Open and Closed Sets in Metric Spaces

A set $S\subseteq \mathbb {R} ^{n}$ is said to be open if $S=\mathrm {int} (S)$, that is, for every point $\mathbf {a} \in S$ we have that there exists a positive real number $r>0$ such that the ball centered at $\mathbf {a}$ with radius $r$ is contained in $S$, i.e., $B(\mathbf {a} ,r)\subseteq S$.

Furthermore, we said that $S\subseteq \mathbb {R} ^{n}$ is closed if $S^{c}$ is open.

For any general metric space $(M,d)$, we define open and closeds subsets $S$ of $M$ in a similar manner.

Definition: If $(M,d)$ is a metric space and $S\subseteq M$ then $S$ is said to be Open if $S=\mathrm {int} (S)$ and $S$ is said to be Closed if $S^{c}$ is open. Moreover, $S$ is said to be Clopen if it is both open and closed.

It is important to note that the definitions above are somewhat of a poor choice of words. A set $S$ may just be open, just closed, open and closed (clopen), or even neither. Unfortunately these definitions are standard and we should note that saying a set is "not open" does not mean it is closed and likewise, saying a set is "not closed" does not mean it is open.

Now consider the whole set $M$. Is $M$ open or closed? Well by definition, for every $a\in M$ there exists a positive real number $r>0$ such that $B(a,r)\subseteq M$ since the ball centered at $a$ with radius $r$ is defined to be the set of all points IN $M$ that are of a distance less than $r$ of $a$. Therefore $M$ is an open set.

So then the complement of $M$ is $M^{c}=M\setminus M=\emptyset$ is a closed set. However, it is vacuously true that for all $a\in \emptyset$ there exists a ball centered at $a$ fully contained in $\emptyset$ since $\emptyset$ contains no points to begin with. Therefore $\emptyset$ is also an open set and so $M$ is also a closed set.

This is the case for all metric spaces $(M,d)$. The whole set $M$ and empty set $\emptyset$ are trivially clopen sets!

Licensing

Content obtained and/or adapted from:

• Open and Closed Balls in Metric Spaces, mathonline.wikidot.com under a CC BY-SA license
• Open and Closed Sets in Metric Spaces, mathonline.wikidot.com under a CC BY-SA license