Families of Sets

Exploration of Mathematical Structures

Set Families

Consider a set \(X\), and a powerset \(\mathcal{P}(X)\) of \(X\). The powerset is a set of all possible subsets of \(X\). The powerset is a lattice, with the partial order \(\subseteq\) as the ordering relation.

In this post, I'll use family to refer to a set containing sets as elements. In general, we can consider a family, but there are other subsets of the powerset. Of particular interest are some of the subsets I've listed below:

Consider a set \(X\), and its powerset \(\mathcal{P}(X)\). We are interested in certain families of subsets of \(X\) that satisfy specific axioms, including:

Use the visualization below to select elements of the powerset and check if they satisfy the axioms of particular subsets of the powerset. To see which elements could be satisfied to satisfy the different axioms, select the corresponding axiom from the dropdown menu, and the elements will be highlighted.

Property Definitions

Contains Whole Set
A family contains the entire base set as a member.

\[X \in \mathcal{A}\]

Contains Empty Set
A family contains the empty set as a member.

\[\emptyset \in \mathcal{A}\]

Closed Under Complements
If a set is in our family, its complement is also in our family.

\[\forall A \in \mathcal{A}: X \setminus A \in \mathcal{A}\]

Closed Under Arbitrary Unions
For any family of sets in our family (even infinitely many), their union is also in our family.

\[\text{For any } \{A_i\}_{i \in I} \subseteq \mathcal{A}: \bigcup_{i \in I} A_i \in \mathcal{A}\]

Closed Under Finite Intersections
For any finite family of sets in our family, their intersection is also in our family.

\[\text{For any } A_1,\ldots,A_n \in \mathcal{A}: \bigcap_{i=1}^n A_i \in \mathcal{A}\]

Closed Under Countable Unions
For any countable family of sets in our family, their union is also in our family.

\[\text{For any } \{A_n\}_{n \in \mathbb{N}} \subseteq \mathcal{A}: \bigcup_{n \in \mathbb{N}} A_n \in \mathcal{A}\]

Downward Directed
A family is downward directed if, for any two sets \(A\) and \(B\) in the family, there exists a set \(C\) in the family such that \(C \subseteq A \cap B\). This implies that the family is closed under directed intersections.

\[\forall A, B \in \mathcal{A}: \exists C \in \mathcal{A}: C \subseteq A \cap B\]

Upward Closed
If a set in our family is contained in another set, that larger set must also be in our family.

\[\forall A \in \mathcal{A}, \forall B \in \mathcal{P}(X): A \subseteq B \implies B \in \mathcal{A}\]

Meet (Greatest Lower Bound)
For any two elements \(a, b \in L\), their greatest lower bound, \(a \wedge b\), satisfies:

Join (Least Upper Bound)
For any two elements \(a, b \in L\), their least upper bound, \(a \vee b\), satisfies:

Mathematical Objects

Axioms of a Topology

  1. \(\emptyset \in \tau\) and \(X \in \tau\) (see Contains Empty Set and Contains Whole Set).
  2. Closed under arbitrary unions (see Closed Under Arbitrary Unions).
  3. Closed under finite intersections (see Closed Under Finite Intersections).

Axioms of a Lattice

  1. Every pair of elements \(a, b \in L\) has a meet.
  2. Every pair of elements \(a, b \in L\) has a join.

Axioms of a \(\sigma\)-Algebra

  1. \(\emptyset \in \Sigma\) (see Contains Empty Set).
  2. Closed under complementation (see Closed Under Complements).
  3. Closed under countable unions (see Closed Under Countable Unions).

Axioms of a Filter

  1. Nonempty.
  2. Closed under finite intersections (see Closed Under Finite Intersections).
  3. Closed under supersets (see Upward Closed).
  4. \(X \in \mathcal{F}\) (see Contains Whole Set).
  5. (Optional) \(\emptyset \notin \mathcal{F}\). I'm not including this axiom in the visualization. If \(\emptyset \in \mathcal{F}\), then by upward closure, all sets are in \(\mathcal{F}\).