Compactness, Local Compactness, Completeness Clearly, every compact metric space is locally compact. I always get confused when completeness is introduced into the picture. Which of the following are true? What are some easy counterexamples to those statements that are false?


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*Every complete metric space is compact.

*Every complete metric space is locally compact.

*Every compact metric space is complete.

*Every locally compact metric space is complete.

*Every locally compact inner product space is of finite dimension.


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*False: the real line with the usual metric is a counterexample.

*False: the irrationals are completely metrizable and nowhere locally compact.

*True: a metric space is compact iff it is complete and totally bounded.

*False: $(0,1)$ with the usual metric is a counterexample.

*True; see this question and this question.
