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On Closed, Totally Bounded Sets

Published online by Cambridge University Press:  20 November 2018

M. G. Murdeshwar
M. G. Murdeshwar*
Affiliation:
University of Alberta, Edmonton
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Extract

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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.

Type
Notes and Problems
Information
Canadian Mathematical Bulletin , Volume 9 , Issue 4 , October 1966 , pp. 525 - 526
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Buck, R. C., ed., Studies in Modern Analysis. The Mathematical Association of America, (1962), page 151.Google Scholar
2. Goffman, C., Real Functions. Rinehart and Co.. New York, (1960), page 63.Google Scholar
3. Pervin, W.J., Foundations of General Topology. Academic Press, New York, (1964), page 127.Google Scholar