On this post:
compact set always contains its supremum and infimum
People ask if compact set always contains its supremum and infimum. I know that it is true on $\mathbb{R}$ with the usual topology.
Pf: Let $K\subseteq \mathbb{R}$ be a nonempty compact set. Suppose not, by Heine Borel $K$ is closed, so the $\sup(K)$ lies in the open set. Take an open ball around the $\sup(K)$, then $\sup(K) -\epsilon$ is the new supremum, contradiction.
However, this requires Heine Borel. And in that post, people are basically saying, that since compactness = closed and bounded, the above proof always works, so compact set always contains supremum and infimum.
Does a compact set in general topological space necessarily contain sup and inf?
In a general topological space there is no order relation - and therefore no meaningful definition of sup and inf.