Field of sets

In mathematics a field of sets is a pair ${\displaystyle \langle X,{\mathcal {F}}\rangle }$ where ${\displaystyle X}$ is a set and ${\displaystyle {\mathcal {F}}}$ is an algebra over ${\displaystyle X}$ i.e., a non-empty subset of the power set of ${\displaystyle X}$ closed under the intersection and union of pairs of sets and under complements of individual sets. In other words, ${\displaystyle {\mathcal {F}}}$ forms a subalgebra of the power set Boolean algebra of ${\displaystyle X}$. (Many authors refer to ${\displaystyle {\mathcal {F}}}$ itself as a field of sets. The word "field" in "field of sets" is not used with the meaning of field from field theory.) Elements of ${\displaystyle X}$ are called points and those of ${\displaystyle {\mathcal {F}}}$ are called complexes and are said to be the admissible sets of ${\displaystyle X}$.

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