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On Closed, Totally Bounded Sets
Published online by Cambridge University Press: 20 November 2018
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C. Goffman asserts that "… in a metric space X a set S is compact if and only if it is closed and totally bounded." [1] and "… every totally bounded sequence in a metric space has convergent subsequence." [2].
The statements (incidentally, equivalent to each other) are both wrong, as the following counter-example shows. Take the set of all reals in the open interval (0, 1) with the usual metric. This space is closed and totally bounded, but not compact.
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