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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>.
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| − | Furthermore, we said that <math>S \subseteq \mathbb{R}^n</math> is closed if <math>S^c</math> is open.
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| − | 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.
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| − | <blockquote style="background: white; border: 1px solid black; padding: 1em;">
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| − | :'''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.
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| − | </blockquote>
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| − | 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.
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| − | 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.
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| − | 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.
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| − | 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!
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| − | ==Licensing==
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| − | Content obtained and/or adapted from:
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| − | * [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
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