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I'm reading Kolmogorov's functional analysis book and I have a question concerning the definition it gives for a compact subset of a metric space (page 51, 1st volume for those interested to check it out).

A set M in the metric space R is said to be compact if every sequence of elements in M contains a subsequence which converges to a point x in R.

I'm confused,shouldn't the subsequence converge to a point x in the set M? In the real numbers a compact set is a bounded and closed one, but the definition above (seems to me) allows the open interval (a,b) to be compact, since every sequence in (a,b) is bounded and therefore contains a convergent subsequence.

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You are correct; $x$ should be required to be in $M$. This is presumably a typo.

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