# Maximum and Minimum of Totally Ordered Compact Sets

Let T be a totally ordered compact set. Does this always imply that a maximum and a minimum element of T exist under this total order?

And if not, what about the special case where:

• T is a closed and bounded interval of the real line (ergo compact) with the usual total order $\leq$ of the reals

If $T$ is a compact set in a linearly ordered space $X$ with its order topology, then $T$ has both a minimum and a maximum element.
To see this, merely note that if $T$ has no largest element, then $\{(\leftarrow,x):x\in T\}$ is an open cover of $T$ with no finite subcover, and if $T$ has no smallest element, then $\{(x,\to):x\in T\}$ is an open cover of $T$ with no finite subcover.
In particular, this is true if $X$ is the real line with the usual topology, since that topology is the one induced by the usual linear order on $\Bbb R$. If the linear order is not related to the topology, however, nothing can be said: it is always possible to put a linear order on an infinite set in such a way that the set has neither a least nor a greatest element.