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 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 and such that and . If is not disconnected then we say that Connected. Furthermore, if 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.
