For any metric space , the following are equivalent (assuming countable choice): is compact. is complete and totally bounded (this is also equivalent to compactness for uniform spaces). is sequentially compact; that is, every sequence in has a convergent subsequence whose limit is in (this is also equivalent to compactness for first-countable uniform spaces). is limit point compact (also called weakly countably compact); that is, every infinite subset of has at least one limit point in . is countably compact; that is, every countable open cover of has a finite subcover. is an image of a continuous function from the Cantor set. Theorem 30.7. Every decreasing nested sequence of nonempty closed subsets in has a nonempty intersection. Every increasing nested sequence of proper open subsets in fails to cover .
For any metric space , the following are equivalent (assuming countable choice): is compact. is complete and totally bounded (this is also equivalent to compactness for uniform spaces). is sequentially compact; that is, every sequence in has a convergent subsequence whose limit is in (this is also equivalent to compactness for first-countable uniform spaces). is limit point compact (also called weakly countably compact); that is, every infinite subset of has at least one limit point in  is countably compact; that is, .every countable open cover of has a finite subcover. is an image of a continuous function from the Cantor set. Theorem 30.7. Every decreasing nested sequence of nonempty closed subsets in has a nonempty intersection. Every increasing nested sequence of proper open subsets in fails to cover .