Connectedness

Connected and Disconnected Metric Spaces

Definition: A metric space $(M,d)$ is said to be Disconnected if there exists nonempty open sets $A$ and $B$ such that $A\cap B=\emptyset$ and $M=A\cup B$. If $M$ is not disconnected then we say that $M$ Connected. Furthermore, if $S\subseteq M$ then $S$ is said to be disconnected/connected if the metric subspace $(S,d)$ is disconnected/connected.

Intuitively, a set is disconnected if it can be separated into two pieces while a set is connected if it’s an entire piece.

For example, consider the metric space $(\mathbb {R} ,d)$ where $d$ is the Euclidean metric on $\mathbb {R}$. Let $S=(a,b)\subset \mathbb {R}$, i.e., $S$ is an open interval in $\mathbb {R}$. We claim that $S$ is connected.

Suppose not. Then there exists nonempty open subsets $A$ and $B$ such that $A\cap B=\emptyset$ and $(a,b)=A\cup B$. Furthermore, $A$ and $B$ must be open intervals themselves, say $A=(c,d)$ and $B=(e,f)$. We must have that $A\cup B=(c,d)\cup (e,f)$. So $c=a$ or $e=a$ and furthermore, $d=b$ or $f=b$.

If $c=a$ then this implies that $f=b$ (since if $d=b$ then $A=(a,b)$ which implies that $B=\emptyset$). So if $A\cup B=(c,d)\cup (e,f)=(a,d)\cup (e,b){\text{ we must have that }}a<d,e<b$. If $d=e$ then $A\cup B=(a,d)\cup (d,b)$ and so $d\not \in (a,b)$ so $A\cup B\neq (a,b)$. If $d<e$ then $A\cup B=(a,d)\cup (e,b)$ and $(d,e)\not \in (a,b)$ so $A\cup B\neq (a,b)$. If $d>e$ then $A\cap B=(e,d)\neq \emptyset$. Either way we see that $(a,b)\neq A\cup B$.

We can use the same logic for the other cases which will completely show that $(a,b)$ is connected.

Basic Theorems Regarding Connected and Disconnected Metric Spaces

A metric space $(M,d)$ is said to be disconnected if there exists $A,B\subseteq M$, $A,B\neq \emptyset$ where $A\cap B=\emptyset$ and:

(1)

${\begin{aligned}\quad M=A\cup B\end{aligned}}$

We say that $(M,d)$ is connected if it is not disconnected.

Furthermore, we say that $S\subseteq M$ is connected/disconnected if the metric subspace $(S,d)$ is connected/disconnected.

We will now look at some important theorems regarding connected and disconnected metric spaces.

Theorem 1: A metric space $(M,d)$ is disconnected if and only if there exists a proper nonempty subset $A\subset M$ such that $A$ is both open and closed.

• $\Rightarrow$ Suppose that $(M,d)$ is disconnected. Then there exists open $A,B\subset M$, $A,B\neq \emptyset$, where $A\cap B=\emptyset$ and $M=A\cup B$.

• Since $A$ is open in $M$ we have that $A^{c}=B$ is closed in $M$. But $B$ is also open. Similarly, since $B$ is open in $M$, $B^{c}=A$ is closed in $M$. So in fact $A$ and $B$ are both nonempty proper subsets of $M$ that are both open and closed.

• $\Leftarrow$ Suppose that there exists a proper nonempty subset $A\subset M$ such that $A$ is both open and closed. Let $B=A^{c}$. Then $B$ is also both open and closed. Furthermore, since $B\neq \emptyset$ and $A\cap B=\emptyset$. Additionally, $M=A\cup B$, so $M$ is disconnected. $\blacksquare$

Theorem 2: If $(M,d)$ is a connected unbounded metric space, then for every $a\in M$ and for all $r>0$, $\{x\in M:d(x,a)=r\}$ is nonempty.

• Proof: Let $(M,d)$ be a connected unbounded metric space and suppose that there exists an $a\in M$ and there exists an $r_{0}>0$ such that:

(2)

${\begin{aligned}\quad \{x\in M:d(x,a)=r_{0}\}=\emptyset \end{aligned}}$

• We will show that a contradiction arises. Let $A=\{x\in M:d(x,a)<r_{0}\}$ and let $B=\{x\in M:d(x,a)>r_{0}\}$. Then $A$ is open since it is simply an open ball centered at $a$. Furthermore, $B$ is open since 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 B^c} is a closed ball centered 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 a} . 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 A} is nonempty since 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 a \in A} and $B$ is nonempty since $(M,d)$ is unbounded (if it were empty then this would imply 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, d)} is bounded). Clearly $A\cap B=\emptyset$ and 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 = A \cup B} . So 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, d)} is a disconnected metric space. But this is a contradiction.

• Therefore the assumption that there exists an 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 a \in M} and an 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 r_0 > 0} such that 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 \{ x \in M : d(x, a) = r_0 \} \emptyset} was false.

• So for all 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 a \in M} and for all 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 r > 0} the 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 \{ x \in M : d(x, a) = r \}} is nonempty. 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 \blacksquare}

Licensing

Content obtained and/or adapted from:

• Connected And Disconnected Metric Spaces, mathonline.wikidot.com under a CC BY-SA license
• Basic Theorems Regarding Connected and Disconnected Metric Spaces, mathonline.wikidot.com under a CC BY-SA license