Definition:Infimum of Set

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

Definition

Let $\struct {S, \preccurlyeq}$ 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 $S$

$(2): \quad d \preccurlyeq c$ 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 $S$) or

$T$ has an infimum (in $S$).

Subset of Real Numbers

The concept is often encountered where $\struct {S, \preccurlyeq}$ 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, $\map \inf T$ 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 refer to the infimum of a set as the meet of the set and use the notation $\bigwedge T$ or $\ds \bigwedge_{y \mathop \in T} y$.

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

Some older sources, applying the concept to a (strictly) decreasing real sequence, refer to an 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

• Definition:Supremum of Set
• Supremum and Infimum are Unique

Special cases

• Definition:Meet (Order Theory)

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): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings
• 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