Definition:Set Ordered by Subset Relation

Definition

Let $S$ be a set.

Then $S$ ordered by the subset relation is the ordered set $\struct {S, \subseteq}$, where $\subseteq$ is the subset relation on $S$.

Ordering by Reverse Inclusion

Let $S$ be ordered by $\supseteq$, the dual of the subset relation $\subseteq$.

Then $S$ is said to be ordered by reverse inclusion.

Also known as

Some sources use the term ordered by inclusion for ordered by the subset relation.

Also see

• Subset Relation is Ordering
• Subset Relation on Power Set is Partial Ordering

Sources

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