nLab
sigma-ideal

σ\sigma-Ideals

Idea

A σ\sigma-ideal is a collection of sets (either subsets of an ambient set or pure sets) that are considered ‘small’ in some fashion. Unlike the notion of ‘small’ in small category, this is not expected to be closed under most infinitary operations, but it is expected to be closed under countably infinitary operations, in particular under countable union.

If we use ‘large’ sets instead, then we have a δ\delta-filter.

Definitions

Let XX be a set. Then a σ\sigma-ideal on XX is a collection \mathcal{I} of subsets of XX such that: 1. If ABA \subset B and BB \in \mathcal{I}, then AA \in \mathcal{I}; 2. If A 1,A 2,A_1, A_2, \ldots \in \mathcal{I}, then there exists BB such that BB \in \mathcal{I} and iA iB\bigcup_i A_i \subseteq B; in light of (1), BB may be assumed to be the union iA i\bigcup_i A_i itself. 3. Some set belongs to \mathcal{I}; in light of (1), the empty set \empty \in \mathcal{I}.

A base of a σ\sigma-ideal is any collection satisfying (2,3); a base is precisely what generates a σ\sigma-ideal by closing under subsets. A subbase of a σ\sigma-ideal is any collection at all; a subbase generates a base by closing under countable unions.

Instead of a set, XX may easily be a proper class; then the elements of \mathcal{I} may be restricted to subclasses that are actually sets. One may take XX to be the class of all pure sets; from the perspective of material set theory, this actually includes the general case above.

As defined above, a σ\sigma-ideal \mathcal{I} is a subset of the power set (or power class) 𝒫X\mathcal{P}X; we can just as easily make \mathcal{I} a subset of any complete lattice \mathcal{L}. Actually, it works just as well if \mathcal{L} if replaced by a sup-semilattice with all countablary suprema, or probably even more generally (see ideal for some idea of how to do that). Note the grammar: a σ\sigma-ideal in LL but on XX (which is the same as in 𝒫X\mathcal{P}X).

Dually, a δ\delta-filter on XX is a collection \mathcal{F} of subsets of XX such that: 1. If ABA \subset B and AA \in \mathcal{F}, then BB \in \mathcal{F}; 2. If A 1,A 2,A_1, A_2, \ldots \in \mathcal{F}, then there exists BB such that BB \in \mathcal{F} and B iA iB \subseteq \bigcap_i A_i; in light of (1), BB may be assumed to be the intersection iA i\bigcap_i A_i itself. 3. Some set belongs to \mathcal{F}; in light of (1), the improper subset XX \in \mathcal{F}.

Using de Morgan duality, δ\delta-filters and σ\sigma-ideals are essentially the same; we have

={AX|¬A} \mathcal{F} = \{ A \subseteq X \;|\; \neg{A} \in \mathcal{I} \}

and vice versa. In constructive mathematics, however, they are not equivalent. Also, the two notions are not equivalent when XX is a proper class.

Examples

  • The power set of XX is both a σ\sigma-ideal and a δ\delta-filter on XX; it is the improper σ\sigma-ideal on XX. (Compare proper ideal.)
  • The null sets in a measure space XX form a σ\sigma-ideal on XX, while the full sets form the corresponding δ\delta-filter.
  • The meagre sets? in a topological space XX form a σ\sigma-ideal on XX, while the comeagre sets form the corresponding δ\delta-filter.
  • The pure sets form a σ\sigma-ideal on (or, equivalently in this case, in) the class of all pure sets; actually, this collection is closed under arbitrary unions rather than merely countablary unions, so we may call it a complete ideal.

Properties

Of course, any σ\sigma-ideal in an ideal. A σ\sigma-ideal is a σ\sigma-ring in its own right. In fact, a σ\sigma-ideal on XX is precisely simultaneously an ideal in and a sub-σ\sigma-ring of 𝒫X\mathcal{P}X. Dual results hold for δ\delta-filters.

Revised on November 13, 2013 19:17:42 by Anonymous Coward (108.72.3.0)