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From various articles in Wikipedia, I have seen some relations between several types of compactness.

In general topological spaces,

Compact $\Longrightarrow$ Sequentially compact $\Longrightarrow$ countably compact $\Longrightarrow$ pseudocompact and weakly countably compact.

I also saw that

  1. In general topological spaces,

    Compact $\Longrightarrow$ σ-compact $\Longrightarrow$ Lindelöf.

    So I was wondering if we can place σ-compact and Lindelöf in the first chain of relations? If yes, what are their positions?

  2. In metric spaces,

    Compact $\Longleftrightarrow$ Sequentially compact $\Longleftrightarrow$ countably compact $\Longleftrightarrow$ pseudocompact $\Longleftrightarrow$ Limit point compact.

    So I was wondering if we can place Limit point compact in the first chain of relations? If yes, what is its position?

Thanks and regards!

  • 1
    This reminded me of this old MO question I asked: http://mathoverflow.net/questions/37195/different-forms-of-compactness-and-their-relation2012-02-01
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    @BrunoStonek: Thanks!2012-02-01

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