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There are many discussions about such question; however, they are proof-type answer

  1. Compactness and sequential compactness in metric spaces
  2. If $(X,d)$ is a metric space then I want to show that limit point compactness and sequential compactness are equivalent.
  3. Compactness and sequential compactness

Could anyone use a simpler description to guide me that when both of them are equivalent and when they are not?

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sleeve chen
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  • I don't understand: are you asking for a simplified proof of this theorem or are you looking for a classification of topological spaces which either are compact both topologically and sequentially or neither topologically nor sequentially compact? –  Jun 13 '16 at 07:13
  • I want a classification. – sleeve chen Jun 13 '16 at 07:17

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