Finite Subsets form Directed Ordering

Theorem

Let $I$ be a set.

Denote with $\FF$ the set of finite subsets of $I$.

Let $\subseteq$ be the subset relation on $\FF$.

Then $\subseteq$ is a directed ordering on $\FF$.

Proof

From Subset Relation is Ordering, we know that $\subseteq$ is an ordering.

Now let $F, G \in \FF$.

From Set Union Preserves Subsets, conclude that $F \cup G \subseteq I$ as $F, G \subseteq I$.

From Union of Finite Sets is Finite, $F \cup G$ is a finite set.

Hence $F \cup G \in \FF$.

Furthermore, $F \subseteq F \cup G$ and $G \subseteq F \cup G$.

It follows that $\subseteq$ is a directed ordering.

$\blacksquare$

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): Appendix $\text{A}$ Preliminaries: $\S 2.$ Topology