Compact Sets in the Order Topology contain their Extrema

Last updated Nov 6, 2022

Let $(X, \leq)$ be a Total Ordering and let $K \subset X$ be Compact. Then $\sup\limits K$ and $\inf\limits K$ both exist. Furthermore, $\sup\limits K \in K$ and $\inf\limits K \in K$.

Recall that A Nonempty Set is Compact in the Order Topology iff it is Tightly Bounded and Complete. Therefore, $K$ has an Upper Bound and a Lower Bound. Because the Order Topology is Hausdorff and Compact Sets in Hausdorff Spaces are Closed, we know $K$ is Closed.