Compact Sets are Bounded

Last updated Nov 6, 2022

Let $(Y, \leq)$ be a Total Ordering with the Order Topology. Let $K \subset Y$ be Compact. Then $K$ is a Bounded Set.

If $K = \emptyset$ then it is vacuously a Bounded Set. Otherwise, if $K$ is Nonempty, then $K$ is a Bounded Set because Nonempty Compact Sets are Tightly Bounded. $\blacksquare$

TODO - the corresponding statement for Metric Spaces

• Empty Set