Definition:Infimum of Set

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

Let $T \subseteq S$.

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


 * $(1): \quad c$ 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 $\struct {\R, \le}$:

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

Also see

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

Special cases

 * Definition:Meet (Order Theory)