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 of the reals

Answer

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 {(,x):xT} is an open cover of T with no finite subcover, and if T has no smallest element, then {(x,):xT} 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 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.

Attribution
Source : Link , Question Author : user19902 , Answer Author : Brian M. Scott

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