# Field of sets

From Encyclopedia of Mathematics

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2020 Mathematics Subject Classification: *Primary:* 03E15 *Secondary:* 28A05 [MSN][ZBL]

Also called Boolean algebra or Algebra of sets. A collection $\mathcal{A}$ of subsets of some set $X$ which contains the empty set and is closed under the set-theoretic operations of finite union, finite intersection and taking complements, i.e. such that

- $A\in\mathcal{A}\Rightarrow X\setminus A\in \mathcal{A}$;
- $A,B\in \mathcal{A}\Rightarrow A\cup B\in\mathcal{A}$;
- $A,B\in \mathcal{A}\Rightarrow A\cap B\in\mathcal{A}$

If a field of sets is also closed under countable unions then it is called $\sigma$-field, or $\sigma$-algebra. We refer the reader to the page Algebra of sets for more on the topic and for several bibliographic references.

**How to Cite This Entry:**

Field of sets.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Field_of_sets&oldid=28889

This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article