Open Sets and Closed Sets in Metric Spaces

Open and Closed Sets in Metric Spaces

A set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} and empty set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \emptyset} are trivially clopen sets!