# Definition:Supremum of Set

*This page is about Supremum in the context of Ordered Set. For other uses, see Supremum.*

## Definition

Let $\struct {S, \preccurlyeq}$ be an ordered set.

Let $T \subseteq S$.

An element $c \in S$ is the **supremum of $T$ in $S$** if and only if:

- $(1): \quad c$ is an upper bound of $T$ in $S$
- $(2): \quad c \preccurlyeq d$ for all upper bounds $d$ of $T$ in $S$.

If there exists a **supremum** of $T$ (in $S$), we say that:

**$T$ admits a supremum (in $S$)**or**$T$ has a supremum (in $S$)**.

### Finite Supremum

If $T$ is finite, $\sup T$ is called a **finite supremum**.

### Subset of Real Numbers

The concept is usually encountered where $\struct {S, \preccurlyeq}$ is the set of real numbers under the usual ordering $\struct {\R, \le}$:

Let $T \subseteq \R$ be a subset of the real numbers.

A real number $c \in \R$ is the **supremum of $T$ in $\R$** if and only if:

- $(1): \quad c$ is an upper bound of $T$ in $\R$
- $(2): \quad c \le d$ for all upper bounds $d$ of $T$ in $\R$.

The **supremum** of $T$ is denoted $\sup T$ or $\map \sup T$.

## Also known as

Particularly in the field of analysis, the supremum of a set $T$ is often referred to as the **least upper bound of $T$** and denoted $\map {\mathrm {lub} } T$ or $\map {\mathrm {l.u.b.} } T$.

Some sources refer to the **supremum of a set** as the **supremum on a set**.

Some sources refer to the **supremum of a set** as the **join of the set** and use the notation $\bigvee T$ or $\ds \bigvee_{y \mathop \in T} y$.

Some sources introduce the notation $\ds \sup_{y \mathop \in T} y$, which may improve clarity in some circumstances.

Some older sources, applying the concept to a (strictly) increasing real sequence, refer to a **supremum** as an **upper limit**.

## Also defined as

Some sources refer to the supremum as being ** the upper bound**.

Using this convention, any element greater than this is not considered to be an upper bound.

## Also see

- Results about
**suprema**can be found**here**.

## Linguistic Note

The plural of **supremum** is **suprema**, although the (incorrect) form **supremums** can occasionally be found if you look hard enough.

## Sources

- 1955: John L. Kelley:
*General Topology*... (previous) ... (next): Chapter $0$: Orderings - 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 14$: Order - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 7$ - 1982: Peter T. Johnstone:
*Stone Spaces*... (previous) ... (next): Chapter $\text I$: Preliminaries, Definition $1.2$ - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 3$: Relations