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I am studying topology using Serge Lang Real Analysis (second edition).In proposition 3.7 the author writes the following :"Compactness implies sequential compactness " and goes on to give a proof that I find convincing . I am confused because the internet gives me the answer that compact doesn't imply sequential compact and gives counter-examples ( that i find a bit difficult to handle ).Can someone explain to me what is going on here is Serge Lang imposing other conditions that i didn't pay attention to or this is a typing mistakes ?

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  • $\begingroup$ Equivalence of various types of compactness is valid only in metric spaces. So there are counterexamples if you go to general topological spaces $\endgroup$ – happymath Mar 12 '17 at 16:46
  • $\begingroup$ Lang probably only considers metric spaces (fairly common for a book about real analysis). For metric spaces, compactness and sequential compactness are equivalent. $\endgroup$ – Daniel Fischer Mar 12 '17 at 16:46
  • $\begingroup$ Lang doesn't consider metric spaces but treats topological spaces in general $\endgroup$ – Mac Sat Mar 12 '17 at 17:21
  • $\begingroup$ math.stackexchange.com/questions/44907/… $\endgroup$ – Zain Patel Mar 12 '17 at 18:48

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