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Let $X$ be a topological space. I know of the following fact:

If $X$ is pseudometrizable or (sequential and Lindelöf, e.g. second-countable) then

$$ X \text{ is compact} \Leftrightarrow X \text{ is countably compact} \Leftrightarrow X \text{ is sequentially compact}. $$

Any of the two above assumptions on $X$ does not imply the other:

  • sequential Lindelöf $\not\Rightarrow$ pseudometrizable (e.g. Helly space)
  • pseudometrizable $\not\Rightarrow$ Lindelöf (e.g. discrete topology on an uncountable set)
  • but we have that pseudometrizable $\Rightarrow$ first-countable $\Rightarrow$ sequential

Question 1: Is there some single known topological property which is implied by any of the two properties "pseudometrizable" or "sequential and Lindelöf" and implies itself the equivalence of the three above compactness properties? More generally, is the equivalence of the three compactness properties known under some (more or less famous) topological property?

Question 2: Is the equivalence of the two compactness properties "compact" and "sequentially compact" known under some (more or less famous) topological property?

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