Difference between revisions of "Connectedness"

Latest revision as of 11:45, 8 November 2021

Definition: A metric space $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M, d)}$ is 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 $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M, d)}$ is said to be disconnected if there exists $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A, B \subseteq M}$ , $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A, B \neq \emptyset}$ where $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \cap B = \emptyset}$ and:

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

We say that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M, d)}$ is connected if it is not disconnected.

Furthermore, we say that $S\subseteq M$ is connected/disconnected if the metric subspace $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (S, d)}$ is connected/disconnected.

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

Theorem 1: A metric space $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M, d)}$ is disconnected if and only if there exists a proper nonempty subset $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \subset M}$ such that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}$ is both open and closed.

$Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Rightarrow}$ Suppose that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M, d)}$ is disconnected. Then there exists open $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A, B \subset M}$ , $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A, B \neq \emptyset}$ , where $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \cap B = \emptyset}$ and $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M = A \cup B}$ .

Since $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}$ is open in $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M}$ we have that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^c = B}$ is closed in $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M}$ . But $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B}$ is also open. Similarly, since $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B}$ is open in $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M}$ , $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B^c = A}$ is closed in $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M}$ . So in fact $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}$ and $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B}$ are both nonempty proper subsets of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M}$ that are both open and closed.

$Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Leftarrow}$ Suppose that there exists a proper nonempty subset $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \subset M}$ such that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}$ is both open and closed. Let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B = A^c}$ . Then $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B}$ is also both open and closed. Furthermore, since $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B \neq \emptyset}$ and $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \cap B = \emptyset}$ . Additionally, $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M = A \cup B}$ , so $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M}$ is disconnected. $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \blacksquare}$

Theorem 2: If $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M, d)}$ is a connected unbounded metric space, then for every $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \in M}$ and for all $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r > 0}$ , $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ x \in M : d(x, a) = r \}}$ is nonempty.

Proof: Let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M, d)}$ be a connected unbounded metric space and suppose 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 there exists an $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_0 > 0}$ 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 \begin{align} \quad \{ x \in M : d(x, a) = r_0 \} = \emptyset \end{align}}

We will show that a contradiction arises. Let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = \{ x \in M : d(x, a) < r_0 \}}$ and let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B = \{ x \in M : d(x, a) > r_0 \}}$ . Then $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}$ is open since it is simply an open ball centered at $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a}$ . Furthermore, $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B}$ 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 $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B}$ is nonempty since $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M, d)}$ 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 $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \cap B = \emptyset}$ 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. $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \blacksquare}$

Continuous Functions on Connected Sets of Metric Spaces

A metric space $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M, d)}$ is said to be disconnected if there exists nonempty open sets $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}$ and $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B}$ such that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \cap B = \emptyset}$ 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 \begin{align} \quad M = A \cup B \end{align}}

If $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M, d)}$ is not disconnected then we say $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M, d)}$ is connected.

Furthermore, we said that a subset $S\subseteq M$ is connected (or disconnected) if the metric subspace $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (S, d)}$ is connected (or disconnected).

We will now look at a nice theorem which tells us that if $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f}$ is continuous on a connected set then the image $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(S)}$ is also connected in the codomain.

Theorem 1: Let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (S, d_S)}$ and $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (T, d_T)}$ be metric spaces, $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C \subseteq S}$ and $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f : C \to T}$ be continuous. If $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C}$ is connected in $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S}$ then $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(C)}$ is connected in $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T}$ .

Proof: Let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}$ be a connected set and suppose that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(C)}$ is not connected, i.e., disconnected. We will show that a contradiction arises.

Suppose that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (f(C), d_T)}$ is disconnected. Then there exists nonempty open sets $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A, B \subset f(C)}$ such that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \cap B = \emptyset}$ 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 \begin{align} \quad f(C) = A \cup B \end{align}}

Since $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f}$ is continuous we have 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 \begin{align} \quad C = f^{-1}(A) \cup f^{-1}(B) \end{align}}

Note that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(A)}$ and $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(B)}$ are nonempty, otherwise, $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}$ or $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B}$ would be empty which cannot happen. Furthermore, both of these sets are open from the continuity of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f}$ . We claim that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1} (A) \cap f^{-1}(B) = \emptyset}$ . Suppose not. Then there exists an $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in f^{-1}(A) \cap f^{-1}(B)}$ and so $f(x)\in A\cap B$ which implies that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \cap B \neq \emptyset}$ which is a contradiction.

Therefore $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(A) \cap f^{-1}(B) = \emptyset}$ and so $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C}$ is a disconnected set. But this is a contradiction.

Hence the assumption that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(C)}$ was disconnected is false. Therefore, if $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C}$ is a connected set in $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S}$ and $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f : C \to T}$ is continuous then $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(C)}$ is connected in $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T}$

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
Continuous Functions on Connected Sets of Metric Spaces, mathonline.wikidot.com under a CC BY-SA license

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 <p>If <span class="math-inline"><math>c = a</math></span> then this implies that <span class="math-inline"><math>f = b</math></span> (since if <span class="math-inline"><math>d = b</math></span> then <span class="math-inline"><math>A = (a, b)</math></span> which implies that <span class="math-inline"><math>B = \emptyset</math></span>). So if <span class="math-inline"><math>A \cup B = (c, d) \cup (e, f) = (a, d) \cup (e, b) \text{ we must have that } a < d, e < b</math></span>. If <span class="math-inline"><math>d = e</math></span> then <span class="math-inline"><math>A \cup B = (a, d) \cup (d, b)</math></span> and so <span class="math-inline"><math>d \not \in (a, b)</math></span> so <span class="math-inline"><math>A \cup B \neq (a, b)</math></span>. If <span class="math-inline"><math>d < e</math></span> then <span class="math-inline"><math>A \cup B = (a, d) \cup (e, b)</math></span> and <span class="math-inline"><math>(d, e) \not \in (a, b)</math></span> so <span class="math-inline"><math>A \cup B \neq (a, b)</math></span>. If <span class="math-inline"><math>d > e</math></span> then <span class="math-inline"><math>A \cap B = (e, d) \neq \emptyset</math></span>. Either way we see that <span class="math-inline"><math>(a, b) \neq A \cup B</math></span>.</p>
 <p>We can use the same logic for the other cases which will completely show that <span class="math-inline"><math>(a, b)</math></span> is connected.</p>
+=== Basic Theorems Regarding Connected and Disconnected Metric Spaces ===
+<p>A metric space <span class="math-inline"><math>(M, d)</math></span> is said to be disconnected if there exists <span class="math-inline"><math>A, B \subseteq M</math></span>, <span class="math-inline"><math>A, B \neq \emptyset</math></span> where <span class="math-inline"><math>A \cap B = \emptyset</math></span> and:</p>
+<div style="text-align: center;"><math>\begin{align} \quad M = A \cup B \end{align}</math></div>
+<p>We say that <span class="math-inline"><math>(M, d)</math></span> is connected if it is not disconnected.</p>
+<p>Furthermore, we say that <span class="math-inline"><math>S \subseteq M</math></span> is connected/disconnected if the metric subspace <span class="math-inline"><math>(S, d)</math></span> is connected/disconnected.</p>
+<p>We will now look at some important theorems regarding connected and disconnected metric spaces.</p>
+<blockquote style="background: white; border: 1px solid black; padding: 1em;">
+<td><strong>Theorem 1:</strong> A metric space <span class="math-inline"><math>(M, d)</math></span> is disconnected if and only if there exists a proper nonempty subset <span class="math-inline"><math>A \subset M</math></span> such that <span class="math-inline"><math>A</math></span> is both open and closed.</td>
+</blockquote>
+<ul>
+<li><span class="math-inline"><math>\Rightarrow</math></span> Suppose that <span class="math-inline"><math>(M, d)</math></span> is disconnected. Then there exists open <span class="math-inline"><math>A, B \subset M</math></span>, <span class="math-inline"><math>A, B \neq \emptyset</math></span>, where <span class="math-inline"><math>A \cap B = \emptyset</math></span> and <span class="math-inline"><math>M = A \cup B</math></span>.</li>
+</ul>
+<ul>
+<li>Since <span class="math-inline"><math>A</math></span> is open in <span class="math-inline"><math>M</math></span> we have that <span class="math-inline"><math>A^c = B</math></span> is closed in <span class="math-inline"><math>M</math></span>. But <span class="math-inline"><math>B</math></span> is also open. Similarly, since <span class="math-inline"><math>B</math></span> is open in <span class="math-inline"><math>M</math></span>, <span class="math-inline"><math>B^c = A</math></span> is closed in <span class="math-inline"><math>M</math></span>. So in fact <span class="math-inline"><math>A</math></span> and <span class="math-inline"><math>B</math></span> are both nonempty proper subsets of <span class="math-inline"><math>M</math></span> that are both open and closed.</li>
+</ul>
+<ul>
+<li><span class="math-inline"><math>\Leftarrow</math></span> Suppose that there exists a proper nonempty subset <span class="math-inline"><math>A \subset M</math></span> such that <span class="math-inline"><math>A</math></span> is both open and closed. Let <span class="math-inline"><math>B = A^c</math></span>. Then <span class="math-inline"><math>B</math></span> is also both open and closed. Furthermore, since <span class="math-inline"><math>B \neq \emptyset</math></span> and <span class="math-inline"><math>A \cap B = \emptyset</math></span>. Additionally, <span class="math-inline"><math>M = A \cup B</math></span>, so <span class="math-inline"><math>M</math></span> is disconnected. <span class="math-inline"><math>\blacksquare</math></span></li>
+</ul>
+<blockquote style="background: white; border: 1px solid black; padding: 1em;">
+<td><strong>Theorem 2:</strong> If <span class="math-inline"><math>(M, d)</math></span> is a connected unbounded metric space, then for every <span class="math-inline"><math>a \in M</math></span> and for all <span class="math-inline"><math>r > 0</math></span>, <span class="math-inline"><math>\{ x \in M : d(x, a) = r \}</math></span> is nonempty.</td>
+</blockquote>
+<ul>
+<li><strong>Proof:</strong> Let <span class="math-inline"><math>(M, d)</math></span> be a connected unbounded metric space and suppose that there exists an <span class="math-inline"><math>a \in M</math></span> and there exists an <span class="math-inline"><math>r_0 > 0</math></span> such that:</li>
+</ul>
+<div style="text-align: center;"><math>\begin{align} \quad \{ x \in M : d(x, a) = r_0 \} = \emptyset \end{align}</math></div>
+<ul>
+<li>We will show that a contradiction arises. Let <span class="math-inline"><math>A = \{ x \in M : d(x, a) < r_0 \}</math></span> and let <span class="math-inline"><math>B = \{ x \in M : d(x, a) > r_0 \}</math></span>. Then <span class="math-inline"><math>A</math></span> is open since it is simply an open ball centered at <span class="math-inline"><math>a</math></span>. Furthermore, <span class="math-inline"><math>B</math></span> is open since <span class="math-inline"><math>B^c</math></span> is a closed ball centered <span class="math-inline"><math>a</math></span>. <span class="math-inline"><math>A</math></span> is nonempty since <span class="math-inline"><math>a \in A</math></span> and <span class="math-inline"><math>B</math></span> is nonempty since <span class="math-inline"><math>(M, d)</math></span> is unbounded (if it were empty then this would imply <span class="math-inline"><math>(M, d)</math></span> is bounded). Clearly <span class="math-inline"><math>A \cap B = \emptyset</math></span> and <span class="math-inline"><math>M = A \cup B</math></span>. So <span class="math-inline"><math>(M, d)</math></span> is a disconnected metric space. But this is a contradiction.</li>
+</ul>
+<ul>
+<li>Therefore the assumption that there exists an <span class="math-inline"><math>a \in M</math></span> and an <span class="math-inline"><math>r_0 > 0</math></span> such that <span class="math-inline"><math>\{ x \in M : d(x, a) = r_0 \} \emptyset</math></span> was false.</li>
+</ul>
+<ul>
+<li>So for all <span class="math-inline"><math>a \in M</math></span> and for all <span class="math-inline"><math>r > 0</math></span> the set <span class="math-inline"><math>\{ x \in M : d(x, a) = r \}</math></span> is nonempty. <span class="math-inline"><math>\blacksquare</math></span></li>
+</ul>
+===Continuous Functions on Connected Sets of Metric Spaces===
+<p>A metric space <span class="math-inline"><math>(M, d)</math></span> is said to be disconnected if there exists nonempty open sets <span class="math-inline"><math>A</math></span> and <span class="math-inline"><math>B</math></span> such that <span class="math-inline"><math>A \cap B = \emptyset</math></span> and:</p>
+<div style="text-align: center;"><math>\begin{align} \quad M = A \cup B \end{align}</math></div>
+<p>If <span class="math-inline"><math>(M, d)</math></span> is not disconnected then we say <span class="math-inline"><math>(M, d)</math></span> is connected.</p>
+<p>Furthermore, we said that a subset <span class="math-inline"><math>S \subseteq M</math></span> is connected (or disconnected) if the metric subspace <span class="math-inline"><math>(S, d)</math></span> is connected (or disconnected).</p>
+<p>We will now look at a nice theorem which tells us that if <span class="math-inline"><math>f</math></span> is continuous on a connected set then the image <span class="math-inline"><math>f(S)</math></span> is also connected in the codomain.</p>
+<blockquote style="background: white; border: 1px solid black; padding: 1em;">
+<td><strong>Theorem 1:</strong> Let <span class="math-inline"><math>(S, d_S)</math></span> and <span class="math-inline"><math>(T, d_T)</math></span> be metric spaces, <span class="math-inline"><math>C \subseteq S</math></span> and <span class="math-inline"><math>f : C \to T</math></span> be continuous. If <span class="math-inline"><math>C</math></span> is connected in <span class="math-inline"><math>S</math></span> then <span class="math-inline"><math>f(C)</math></span> is connected in <span class="math-inline"><math>T</math></span>.</td>
+</blockquote>
+<ul>
+<li><strong>Proof:</strong> Let <span class="math-inline"><math>A</math></span> be a connected set and suppose that <span class="math-inline"><math>f(C)</math></span> is not connected, i.e., disconnected. We will show that a contradiction arises.</li>
+</ul>
+<ul>
+<li>Suppose that <span class="math-inline"><math>(f(C), d_T)</math></span> is disconnected. Then there exists nonempty open sets <span class="math-inline"><math>A, B \subset f(C)</math></span> such that <span class="math-inline"><math>A \cap B = \emptyset</math></span> and:</li>
+</ul>
+<div style="text-align: center;"><math>\begin{align} \quad f(C) = A \cup B \end{align}</math></div>
+<ul>
+<li>Since <span class="math-inline"><math>f</math></span> is continuous we have that:</li>
+</ul>
+<div style="text-align: center;"><math>\begin{align} \quad C = f^{-1}(A) \cup f^{-1}(B) \end{align}</math></div>
+<ul>
+<li>Note that <span class="math-inline"><math>f^{-1}(A)</math></span> and <span class="math-inline"><math>f^{-1}(B)</math></span> are nonempty, otherwise, <span class="math-inline"><math>A</math></span> or <span class="math-inline"><math>B</math></span> would be empty which cannot happen. Furthermore, both of these sets are open from the continuity of <span class="math-inline"><math>f</math></span>. We claim that <span class="math-inline"><math>f^{-1} (A) \cap f^{-1}(B) = \emptyset</math></span>. Suppose not. Then there exists an <span class="math-inline"><math>x \in f^{-1}(A) \cap f^{-1}(B)</math></span> and so <span class="math-inline"><math>f(x) \in A \cap B</math></span> which implies that <span class="math-inline"><math>A \cap B \neq \emptyset</math></span> which is a contradiction.</li>
+</ul>
+<ul>
+<li>Therefore <span class="math-inline"><math>f^{-1}(A) \cap f^{-1}(B) = \emptyset</math></span> and so <span class="math-inline"><math>C</math></span> is a disconnected set. But this is a contradiction.</li>
+</ul>
+<ul>
+<li>Hence the assumption that <span class="math-inline"><math>f(C)</math></span> was disconnected is false. Therefore, if <span class="math-inline"><math>C</math></span> is a connected set in <span class="math-inline"><math>S</math></span> and <span class="math-inline"><math>f : C \to T</math></span> is continuous then <span class="math-inline"><math>f(C)</math></span> is connected in <span class="math-inline"><math>T</math></span></li>
+</ul>
 == Licensing ==
 Content obtained and/or adapted from:
 * [http://mathonline.wikidot.com/connected-and-disconnected-metric-spaces Connected And Disconnected Metric Spaces, mathonline.wikidot.com] under a CC BY-SA license
+* [http://mathonline.wikidot.com/basic-theorems-regarding-connected-and-disconnected-metric-s Basic Theorems Regarding Connected and Disconnected Metric Spaces, mathonline.wikidot.com] under a CC BY-SA license
+* [http://mathonline.wikidot.com/continuous-functions-on-connected-sets-of-metric-spaces Continuous Functions on Connected Sets of Metric Spaces, mathonline.wikidot.com] under a CC BY-SA license

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