Definition:Complete Lattice/Definition 2

Definition

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

Then $\struct {S, \preceq}$ is a complete lattice if and only if:

$\forall S' \subseteq S: \inf S', \sup S' \in S$

That is, if and only if all subsets of $S$ have both a supremum and an infimum.

Also see

• Equivalence of Definitions of Complete Lattice

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $39$. Order Topology: $7$