# Definition:Infimum of Set

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

## Definition

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

Let $T \subseteq S$.

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

- $(1): \quad c$ is a lower bound of $T$ in $S$
- $(2): \quad d \preceq c$ for all lower bounds $d$ of $T$ in $S$.

If there exists an **infimum** of $T$ (in $S$), we say that **$T$ admits an infimum (in $S$)**.

### Subset of Real Numbers

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

Let $T \subseteq \R$.

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

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

The **infimum** of $T$ is denoted $\inf T$ or $\map \inf T$.

### Finite Infimum

If $T$ is finite, $\inf T$ is called a **finite infimum**.

## Also known as

Particularly in the field of analysis, the infimum of a set $T$ is often referred to as the **greatest lower bound of $T$** and denoted $\map {\mathrm {glb} } T$ or $\map {\mathrm {g.l.b.} } T$.

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

Some sources introduce the notation $\displaystyle \inf_{y \mathop \in S} y$, which may improve clarity in some circumstances.

Some older sources, applying the concept to a (strictly) decreasing real sequence, refer to a **infimum** as a **lower limit**.

## Also defined as

Some sources refer to the infimum as being ** the lower bound**.

Using this convention, any element less than this is not considered to be a lower bound.

## Also see

### Special cases

## Linguistic Note

The plural of **infimum** is **infima**, although the (incorrect) form **infimums** 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): $\S 14$ - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 7$ - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 3$: Relations