Definition:Infimum of Set

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

Let $T \subseteq S$.

An element $c \in S$ is the infimum of $T$ in $S$ :


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

The infimum of $T$ is denoted $\inf T$.

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 $\left({S, \preceq}\right)$ is the set of real numbers under the usual ordering: $\left({\R, \le}\right)$:

Also known as
The infimum of $T$ is often called the greatest lower bound of $T$ and denoted $\operatorname{glb} \left({T}\right)$.

Some sources refer to the infimum as being the lower bound. Using this convention, any number smaller than this is not considered to be a lower bound.

Linguistic Note
The plural of infimum is infima, although the (incorrect) form infimums can occasionally be found if you look hard enough.

Also see

 * Definition:Supremum of Set


 * Supremum and Infimum are Unique