A set $A$ with an order relation $<$ is said to be well-ordered if every nonempty subset of $A$ has a Smallest element.

Well-ordering theorem

Example of a Hausdorff space which is limit point compact but not compact (Proof here) – Theorem. There is an uncoutnable Well-ordered set such that every section $(-\infty, a)$ is countable. ("the minimal well-ordered uncountable set"?). Uncountable well-ordered sets are related to Transfinite numbers. Corollary S_Omega is not metrizable, like $\bar{S_\Omega}$.. However, S_Omega is first countable but not first countable, but not second countable