Bases of Open Sets - Revision history

Khanh at 23:49, 13 November 2021

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Khanh at 23:28, 13 November 2021

2021-11-13T23:28:47Z

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Khanh: Created page with "A set $S \subseteq \mathbb{R}^n$ is said to be open if $S =\mathrm{int} (S)$ , that is, for every point $\mathbf{a} \in S$ we have that there e..."

2021-11-13T23:26:15Z

Created page with "A set <math>S \subseteq \mathbb{R}^n</math> is said to be open if <math>S =\mathrm{int} (S)</math>, that is, for every point <math>\mathbf{a} \in S</math> we have that there e..."

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A set <math>S \subseteq \mathbb{R}^n</math> is said to be open if <math>S =\mathrm{int} (S)</math>, that is, for every point <math>\mathbf{a} \in S</math> we have that there exists a positive real number <span class="math-inline"><math>r > 0</math></span> such that the ball centered at <math>\mathbf{a}</math> with radius <math>r</math> is contained in <math>S</math>, i.e., <math>B(\mathbf{a}, r) \subseteq S</math>.

Furthermore, we said that <math>S \subseteq \mathbb{R}^n</math> is closed if <math>S^c</math> is open.

For any general metric space <math>(M, d)</math>, we define open and closeds subsets <math>S</math> of <math>M</math> in a similar manner.

<blockquote style="background: white; border: 1px solid black; padding: 1em;">
:'''Definition:''' If <math>(M, d)</math> is a metric space and <math>S \subseteq M</math> then <math>S</math> is said to be '''Open''' if <math>S = \mathrm{int} (S)</math> and <math>S</math> is said to be '''Closed<''' if <math>S^c</math> is open. Moreover, <math>S</math> is said to be '''Clopen''' if it is both open and closed.
</blockquote>

It is important to note that the definitions above are somewhat of a poor choice of words. A set <math>S</math> 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 <math>M</math>. Is <math>M</math> open or closed? Well by definition, for every <math>a \in M</math> there exists a positive real number <math>r > 0</math> such that <math>B(a, r) \subseteq M</math> since the ball centered at <math>a</math> with radius <math>r</math> is defined to be the set of all points <em>IN <math>M</math></em> that are of a distance less than <math>r</math> of <math>a</math>. Therefore <math>M</math> is an open set.

So then the complement of <math>M</math> is <math>M^c = M \setminus M = \emptyset</math> is a closed set. However, it is vacuously true that for all <math>a \in \emptyset</math> there exists a ball centered at <math>a</math> fully contained in <math>\emptyset</math> since <math>\emptyset</math> contains no points to begin with. Therefore <math>\emptyset</math> is also an open set and so <math>M</math> is also a closed set.

This is the case for all metric spaces <math>(M, d)</math>. The whole set <math>M</math> and empty set <math>\emptyset</math> are trivially clopen sets!

==Licensing==
Content obtained and/or adapted from:
* [http://mathonline.wikidot.com/open-and-closed-sets-in-metric-spaces Open and Closed Sets in Metric Spaces, mathonline.wikidot.com] under a CC BY-SA license

@@ Line 1: / Line 1: @@
 == Licensing ==
 Content obtained and/or adapted from:
 * [http://mathonline.wikidot.com/bases-of-a-topology Bases of a topology, mathonline.wikidot.com] under a CC BY-SA license

← Older revision	Revision as of 23:28, 13 November 2021
Line 1:	Line 1:
−	+	== Licensing ==
	+	Content obtained and/or adapted from:
	+	* [http://mathonline.wikidot.com/bases-of-a-topology Bases of a topology, mathonline.wikidot.com] under a CC BY-SA license

← Older revision		Revision as of 23:27, 13 November 2021
Line 1:		Line 1:
−	A set <math>S \subseteq \mathbb{R}^n</math> is said to be open if <math>S =\mathrm{int} (S)</math>, that is, for every point <math>\mathbf{a} \in S</math> we have that there exists a positive real number <span class="math-inline"><math>r > 0</math></span> such that the ball centered at <math>\mathbf{a}</math> with radius <math>r</math> is contained in <math>S</math>, i.e., <math>B(\mathbf{a}, r) \subseteq S</math>.

−	~~Furthermore, we said that <math>S \subseteq \mathbb{R}^n</math> is closed if <math>S^c</math> is open.~~
−
−	~~For any general metric space <math>(M, d)</math>, we define open and closeds subsets <math>S</math> of <math>M</math> in a similar manner.~~
−
−	~~<blockquote style="background: white; border: 1px solid black; padding: 1em;">~~
−	:'''Definition:''' If <math>(M, d)</math> is a metric space and <math>S \subseteq M</math> then <math>S</math> is said to be '''Open''' if <math>S = \mathrm{int} (S)</math> and <math>S</math> is said to be '''Closed<''' if <math>S^c</math> is open. Moreover, <math>S</math> is said to be '''Clopen''' if it is both open and closed.
−	~~</blockquote>~~
−
−	It is important to note that the definitions above are somewhat of a poor choice of words. A set <math>S</math> 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 <math>M</math>. Is <math>M</math> open or closed? Well by definition, for every <math>a \in M</math> there exists a positive real number <math>r > 0</math> such that <math>B(a, r) \subseteq M</math> since the ball centered at <math>a</math> with radius <math>r</math> is defined to be the set of all points <em>IN <math>M</math></em> that are of a distance less than <math>r</math> of <math>a</math>. Therefore <math>M</math> is an open set.
−
−	So then the complement of <math>M</math> is <math>M^c = M \setminus M = \emptyset</math> is a closed set. However, it is vacuously true that for all <math>a \in \emptyset</math> there exists a ball centered at <math>a</math> fully contained in <math>\emptyset</math> since <math>\emptyset</math> contains no points to begin with. Therefore <math>\emptyset</math> is also an open set and so <math>M</math> is also a closed set.
−
−	~~This is the case for all metric spaces <math>(M, d)</math>. The whole set <math>M</math> and empty set <math>\emptyset</math> are trivially clopen sets!~~
−
−
−	~~==Licensing==~~
−	~~Content obtained and/or adapted from:~~
−	* [http://mathonline.wikidot.com/open-and-closed-sets-in-metric-spaces Open and Closed Sets in Metric Spaces, mathonline.wikidot.com] under a CC BY-SA license

Bases of Open Sets - Revision history

Khanh at 23:49, 13 November 2021

Khanh at 23:28, 13 November 2021

Khanh: Blanked the page

Khanh: Created page with "A set 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 e..."

Khanh: Created page with "A set $S \subseteq \mathbb{R}^n$ is said to be open if $S =\mathrm{int} (S)$ , that is, for every point $\mathbf{a} \in S$ we have that there e..."