Definition 6

Let $E$ be an ordered set and let $X$ be a subset of $E$. An element of $E$ is said to be the greatest lower bound or infimum (resp. least upper bound or supremum) of $X$ in $E$ if it is the greatest (resp. least) element of the set of lower (resp. upper) bounds of $X$ in $E$.

Given a subset $X$ of an ordered set $E$, the least upper bound (resp. greatest lower bound) of $X$ in $E$, when it exists, is denoted by

or by sup X (resp. inf X) if there is no risk of ambiguity. The least upper bound (resp. greatest lower bound) of a set $\{x, y\}$ of two elements, when it exists, is denoted by sup $(x, y)$ (resp. inf $(x, y)$). Similarly for the least upper bound and greatest lower bound of a set of three elements, etc.

¶ If a subset X of E has a greatest element $a$, then $a$ is the least upper bound of X in E.

¶ If X has a greatest lower bound $a$ in E, then $a$ is the least upper bound of X with respect to the opposite ordering on E. For this reason we may restrict ourselves for the most part, in what follows, to consideration of the properties of least upper bounds.

Examples

(1) The set of upper bounds of the empty set $\emptyset$ in an ordered set E is evidently E itself; hence $\emptyset$ has a supremum in E if and only if E has a least element, which is then the least upper bound of $\emptyset$.

(2) In the set $\mathfrak{P}(E)$ of subsets of a set E, ordered by inclusion, every subset $\mathfrak{S}$ of $\mathfrak{P}(E)$ has a least upper bound, namely the union of the sets of $\mathfrak{S}$, and a greatest lower bound, namely the intersection of the sets of $\mathfrak{S}$.

(3) Let E, F be two sets and let $\Theta$ be a subset of the $\Phi(E, F)$ of mappings of subsets of E into F, ordered by extension of mappings (no. 1, Example 3). For each $u \in \Phi(E, F)$ let $D(u)$ be the domain of $u$. The condition for the existence of a common extension of a family of mappings belonging to $\Phi(E, F)$ (Chapter II, § 4, no. 6, Proposition 7) shows that $\Theta$ has a least upper bound in $\Phi(E, F)$ if and only if for each pair $(u, v)$ of elements of $\Theta$ we have $u(x) = v(x)$ whenever $x \in D(u) \cap D(v)$.

A mapping $f$ of a set A into an ordered set E is said to have a least upper bound if the image $f(A)$ has a least upper bound in E; this bound is then called the least upper bound of f and is written $\sup_{x \in A} f(x)$. Similarly for the greatest lower bound.

In particular, if a subset A of E has a least upper bound in E, this bound is the least upper bound of the canonical injection of A into E, and may therefore be written as $\sup_{x \in A} x$.