Is Every Sequentially Compact, First Countable Space Also Compact? I want to know if sequential compactness in first countable space implies compactness. I know the implication is correct in metric space (a proof that uses Lebesgue number), but I cannot mimic a proof in a general first countable space.
Or perhaps we cannot get compactness when we only have sequential compactness, but only countable compactness or Lindelof?

Sequential compactness by itself implies countable compactness, as proven on ProofWiki.
$\pi$-Base suggests that the Open Ordinal Space $[0, \Omega)$ and the Long Line are both sequentially compact and first countable, yet neither is compact or even Lindelof.
