# Definition:Upper Set

## Definition

Let $\left({S, \preceq}\right)$ be an ordered set.

Let $U \subseteq S$.

### Definition 1

$U$ is an upper set in $S$ if and only if:

$\forall u \in U: \forall s \in S: u \preceq s \implies s \in U$

### Definition 2

$U$ is an upper set in $S$ if and only if:

$U^\succeq \subseteq U$

where $U^\succeq$ is the upper closure of $U$.

### Definition 3

$U$ is an upper set in $S$ if and only if:

$U^\succeq = U$

where $U^\succeq$ is the upper closure of $U$.

## Also known as

An upper set is also known as an upper-closed set or upward-closed set. Some sources call it an upset or up-set.

## Also see

• Results about upper sets can be found here.