Compact space

Limit point compact

Sequentially compact

Countably compact

X compact implies X limit point compact

Lemma: F metric and limit point compact, then it is sequentially compact.

Lebesgue number lemma

Theorem. Let $X$ be Metrizable, then The following are equivalent:

1. $X$ is compact
2. $X$ is limit point compact
3. $X$ is sequentially compact

Proof

1. => 2. (generally true)

2. => 3. (previous lemma, which needs metric)

remains 3. => 1.

use Lemma: Suppose X is metric and sequentially compact, then for each $\epsilon > 0$, there is a finite covering of $X$ by open $\epsilon$-balls, and also the Lebesgue number lemma

Example of a Hausdorff space which is limit point compact but not compact (Proof here), $S_\Omega$ – Theorem. There is an uncoutnable Well-ordered set such that every section $(-\infty, a)$ is countable. ("the minimal well-ordered uncountable set"?). Infinite 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