Ideaalsetest

Ideaalsetest is a term used in Estonian mathematical literature to refer to the collection of subsets of a given set that form an ideal. An ideaalset (plural ideaalset) on a base set X is a nonempty family I of subsets of X that satisfies two key properties: downward closure (if A is in I and B is a subset of A, then B is also in I) and closure under finite unions (if A and B are in I, then A ∪ B is in I). In many treatments, an ideal must not contain the whole set X.

Common examples illustrate the concept. The finite-sets ideal Fin(X) consists of all finite subsets of an infinite X. The null ideal on a measure space contains all subsets of measure zero. In topology, the family of meager sets forms a σ-ideal, closed under countable unions. Ideaalsetest are thus used to formalize what is considered “small” or negligible within a given context.

Relation to other concepts is important. The dual notion to an ideal is a filter, which is closed under supersets and finite intersections and contains the whole base set. Ideaalsetest can be combined with other structures, such as sigma-algebras, to study measure, category, and related notions. In set theory and analysis, ideals play a central role in describing convergence, genericity, and decomposition of spaces.

See also: ideal (set theory), σ-ideal, filter, Boolean algebra.