Field of Sets
- is a mathematical structure consisting of a pair (π,πΉ) where:
- π is a set
- πΉ is a family of subsets of π called an algebra over π following the conditions:
- πΉ contains the empty set as an element; β
βπΉ
- πΉ is closed under the complement of π; if {π₯}βπΉ, then π\{π₯}βπΉ
- πΉ is closed under finite unions; if {π₯}βπΉ and {π¦}βπΉ, then {π₯}β{π¦}βπΉ
- πΉ is closed under finite intersections; if {π₯}βπΉ and {π¦}βπΉ, then {π₯}β{π¦}βπΉ
- field of sets play an essential role in the representation theory of Boolean algebra. Every Boolean algebra can be represented as a field of sets
- if πΉ is also closed under countable unions and/or intersections, then πΉ is also aΒ Ο-algebra