Ideaalset

Ideaalset is a mathematical concept referring to an ideal of subsets on a base set. An ideal I on a set X is a nonempty family of subsets of X that is closed under taking subsets and under finite unions. Concretely, ∅ belongs to I; if A ⊆ X and A ∈ I, then any B ⊆ A also lies in I; and if A, B ∈ I, then A ∪ B ∈ I. Often I is required to be proper, meaning I ≠ P(X), so there exist subsets of X not in I.

Ideaalsets are used to formalize the notion of “small” or “negligible” subsets of X. The dual notion is a filter on X, defined by F = {X \ A : A ∈ I}, which consists of the “large” or “co-small” sets. If I is the ideal of finite subsets of X, then the corresponding filter consists of cofinite sets.

Common examples include:

- The finite sets on X, Fin(X). This is the smallest nontrivial ideal on X.

- The ideal of measure-zero sets with respect to a given measure, forming a sigma-ideal (closed under countable unions) in a measure space.

- The ideal of nowhere dense or meager sets in a topological space, which is a sigma-ideal in many contexts.

Related notions include sigma-ideals (ideals closed under countable unions), principal ideals (generated by a particular family of subsets), and ultrafilters (maximal filters). Ideaalsets play a central role in set theory, topology, measure theory, and the study of cardinal invariants, providing a rigorous framework for discussing size and negligibility of subsets.