Algebra: Difference between revisions
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Let <math>X</math> be a nonempty set. An '''algebra''' <math>\mathcal{A} \subseteq 2^X</math> is a nonempty collection of subsets of <math>X</math> that is closed under finite unions and complements. | Let <math>X</math> be a nonempty set. An '''algebra'''<ref name="Folland"/> <math>\mathcal{A} \subseteq 2^X</math> is a nonempty collection of subsets of <math>X</math> that is closed under finite unions and complements. | ||
By DeMorgan's laws, an algebra is also closed under finite intersections, and also contains the empty set and <math>X</math> itself. | By DeMorgan's laws, an algebra is also closed under finite intersections, and also contains the empty set and <math>X</math> itself. | ||
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==Non-examples== | ==Non-examples== | ||
* Given a topological space <math>(X,\tau)</math>, the topology <math>\tau</math> is in general not an algebra, for the complement of an open set <math>U \in \tau</math> may fail to be open. For example, in <math>X = \mathbb{R}</math> with the standard topology, the open interval <math>(0,1)</math> is open, but its complement <math>(0,1)^c = (-\infty,0] \cup [1,+\infty)</math> is not. | * Given a topological space <math>(X,\tau)</math>, the topology <math>\tau</math> is in general not an algebra, for the complement of an open set <math>U \in \tau</math> may fail to be open. For example, in <math>X = \mathbb{R}</math> with the standard topology, the open interval <math>(0,1)</math> is open, but its complement <math>(0,1)^c = (-\infty,0] \cup [1,+\infty)</math> is not. | ||
==References== | |||
<references> | |||
(Still figuring out how to get the citation to work...) | |||
<ref name="Folland" >{{citation | last=Folland | first=Gerald B.| title=Real Analysis: Modern Techniques and Their Applications | year = 1999 | publisher = John Wiley and Sons | isbn=0471317160 }} Second edition.</ref> | |||
</references> |
Revision as of 17:58, 26 October 2020
Let be a nonempty set. An algebra[1] is a nonempty collection of subsets of that is closed under finite unions and complements.
By DeMorgan's laws, an algebra is also closed under finite intersections, and also contains the empty set and itself.
Examples of -algebras
Assume that is nonempty.
- Given a set , then and are algebras.
- Given a set , the collection of all finite and cofinite (having finite complement) subsets of is an algebra.
- A sigma-algebra is a particular example of an algebra.
Non-examples
- Given a topological space , the topology is in general not an algebra, for the complement of an open set may fail to be open. For example, in with the standard topology, the open interval is open, but its complement is not.
References
- ↑ Template:Citation Second edition.