# Definition:Set Ordered by Subset Relation

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## 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

## Sources

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