The infimum of a set $S$ is the greatest lower bound of $S$ and is denoted $\inf(S)$

Let $A$ be a set with a partial order $\leq$ and let $S \subseteq A$ For any $x \in A$ $x$ is a lower bound of $S$ if $x \leq y$ for any $y \in S$ The infimum of $S$ denoted $\inf(S)$ is the greatest such lower bound; that is, if $b$ is a lower bound of $S$ then $b \leq \inf(S)$

Note that it is not necessarily the case that $\inf(S) \in S$ Suppose $S = (0, 1)$ then $\inf(S) = 0$ but $0 \not\in S$

Also note that a set does not necessarily have an infimum. See the attachments to this entry for examples.