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I know that compact and sequentially compact are usually not related, i.e. one does not imply the other, for general topological space.

However, I saw a theorem (in Folland's real analysis) saying that in general topological space the following statements are equivalent:

  1. $X$ is compact.

  2. Every net in $X$ has a cluster point.

  3. Every net in $X$ has a convergent subnet.

So how is statement 3 different from sequentially compact?

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  • $\begingroup$ See, for instance, this question. Or you could take a look at the related questions to the left (although now, since I've linked to them, they will appear under "linked" instead) and find this one or this one or many others. $\endgroup$
    – Arthur
    Commented Nov 20, 2017 at 20:32
  • $\begingroup$ @Arthur Okay the second and the third links help. I guess I didn't realize the (huge) difference between a sequence and a net. $\endgroup$
    – k99731
    Commented Nov 20, 2017 at 20:42
  • $\begingroup$ Sequences can only be countable. That makes a huge difference in general. $\endgroup$
    – Arthur
    Commented Nov 20, 2017 at 20:46

1 Answer 1

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"Sequentially compact" means "every sequence has a convergent subsequence". Since a sequence is a very special kind of net (namely one with index set $\mathbb{N}$), this is not the same as statement (3). It is neither weaker nor stronger than statement (3), because on the one hand you only consider "every sequence" rather than "every net" (making it weaker), but on the other hand you are required to have a subsequence rather than just a subnet (making it stronger).

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