ideal (family of subsets)

Definition

Given a nonempty set X, a nonempty family I of subsets in X is called an ideal on X iff

1. (finite additivity) if J and K are in I, then the union of J and K is also in I;
2. (heredity) if J is in I and K is contained in J, then K is also in I.

For a subset A of X, the family of all subsets of A is called the principal ideal of A, denoted by <A>.

Remark

• Important examples: finite subsets, countable subsets, nowhere dense subsets, meager (first category) subsets, scattered subsets (in a TI space), and subsets of measure zero in a complete measure space.
• See I-compact (compact modulo I), e-compact, e-paracompact.

Reference

• T. R. Hamlett and Dragan Jankovic, On Weaker Forms of Paracompactness, Countable Compactness, and Lindelöfness, Annals of the New York Academy of Sciences Volume 728, General Topology and Applications pages 41–49, November 1994.