Difference between revisions of "Connectedness"
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<p>For example, consider the metric space <span class="math-inline"><math>(\mathbb{R}, d)</math></span> where <span class="math-inline"><math>d</math></span> is the Euclidean metric on <span class="math-inline"><math>\mathbb{R}</math></span>. Let <span class="math-inline"><math>S = (a, b) \subset \mathbb{R}</math></span>, i.e., <span class="math-inline"><math>S</math></span> is an open interval in <span class="math-inline"><math>\mathbb{R}</math></span>. We claim that <span class="math-inline"><math>S</math></span> is connected.</p> | <p>For example, consider the metric space <span class="math-inline"><math>(\mathbb{R}, d)</math></span> where <span class="math-inline"><math>d</math></span> is the Euclidean metric on <span class="math-inline"><math>\mathbb{R}</math></span>. Let <span class="math-inline"><math>S = (a, b) \subset \mathbb{R}</math></span>, i.e., <span class="math-inline"><math>S</math></span> is an open interval in <span class="math-inline"><math>\mathbb{R}</math></span>. We claim that <span class="math-inline"><math>S</math></span> is connected.</p> | ||
<p>Suppose not. Then there exists nonempty open subsets <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 <span class="math-inline"><math>(a, b) = A \cup B</math></span>. Furthermore, <span class="math-inline"><math>A</math></span> and <span class="math-inline"><math>B</math></span> must be open intervals themselves, say <span class="math-inline"><math>A = (c, d)</math></span> and <span class="math-inline"><math>B = (e, f)</math></span>. We must have that <span class="math-inline"><math>A \cup B = (c, d) \cup (e, f)</math></span>. So <span class="math-inline"><math>c = a</math></span> or <span class="math-inline"><math>e = a</math></span> and furthermore, <span class="math-inline"><math>d = b</math></span> or <span class="math-inline"><math>f = b</math></span>.</p> | <p>Suppose not. Then there exists nonempty open subsets <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 <span class="math-inline"><math>(a, b) = A \cup B</math></span>. Furthermore, <span class="math-inline"><math>A</math></span> and <span class="math-inline"><math>B</math></span> must be open intervals themselves, say <span class="math-inline"><math>A = (c, d)</math></span> and <span class="math-inline"><math>B = (e, f)</math></span>. We must have that <span class="math-inline"><math>A \cup B = (c, d) \cup (e, f)</math></span>. So <span class="math-inline"><math>c = a</math></span> or <span class="math-inline"><math>e = a</math></span> and furthermore, <span class="math-inline"><math>d = b</math></span> or <span class="math-inline"><math>f = b</math></span>.</p> | ||
| − | <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) | + | <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> | <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> | ||
Revision as of 11:10, 8 November 2021
Connected and Disconnected Metric Spaces
Definition: A metric space is said to be Disconnected if there exists nonempty open sets and such that 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} . 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} is not disconnected then 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} Connected. Furthermore, 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 S \subseteq M} 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 S} is said to be disconnected/connected 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 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 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 (\mathbb{R}, d)} 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 d} is the Euclidean metric on 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 \mathbb{R}} . 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 = (a, b) \subset \mathbb{R}} , i.e., 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} is an open interval 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 \mathbb{R}} . 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 S} is connected.
Suppose not. Then there exists nonempty open subsets 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 (a, b) = A \cup B} . 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 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} must be open intervals themselves, 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 A = (c, d)} 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 = (e, f)} . We must 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 \cup B = (c, d) \cup (e, f)} . 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 = 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 e = a} and 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 d = b} 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 f = b} .
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 = a} then this 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 f = b} (since 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 d = b} 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 = (a, 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 B = \emptyset} ). So 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 A \cup B = (c, d) \cup (e, f) = (a, d) \cup (e, b) \text{ we must have that } a < d, e < b} . 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 d = e} 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 \cup B = (a, d) \cup (d, b)} 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 d \not \in (a, 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 A \cup B \neq = (a, b)} . 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 d < e} 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 \cup B = (a, d) \cup (e, b)} 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 (d, e) \not \in (a, 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 A \cup B \neq (a, b)} . 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 d > e} 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 \cap B = (e, d) \neq \emptyset} . Either way we see 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, b) \neq A \cup B} .
We can use the same logic for the other cases which will completely show 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, b)} is connected.
