Upper bound

An element u with s≤u for every s in a subset S of a poset

Upper bound

Let $(P,\le)$ be a partially ordered set and let $S\subseteq P$ be a subset . An element $u\in P$ is an upper bound of $S$ if

\forall s\in S,\; s\le u.

Upper bounds are used to formulate maximality principles such as Zorn's lemma . In ordered structures like $\mathbb{R}$ , one often asks whether a least upper bound exists (the supremum ).

Examples:

In $(\mathbb{R},\le)$ , the number $1$ is an upper bound of $(0,1)$ .
In $(\mathcal{P}(X),\subseteq)$ , the union of a family of subsets is an upper bound of that family.
In $(\mathbb{Z},\le)$ , the set $\{n\in\mathbb{Z}: n\le 0\}$ has no upper bound.