Least Upper Bound Property

Definition

Consider an Partially Ordered Set $X$. The set $X$ has the least upper bound property if every bounded-above non-empty subset $Y$ of $X$ has the least Upper Bound.

Examples

real number $\mathbb{R}$ with usual order has the least upper bound property. rational numbers $\mathbb{R}$ with usual order does not have the least upper bound property ($\because (0,\sqrt{2})\cap \mathbb{Q}$ has no Supremum in $\mathbb{Q}$).

Facts

Consider a Totally Ordered Set $X$ having the least upper bound property. Then, in the Order Topology defined on $X$, any closed interval (closed and bounded) in $X$ is compact. (It’s a generalization of Heine-Borel Theorem)