（原文） Metric spaces For any metric space (X, d), the following are equivalent (assuming countable choice): 1.(X, d) is compact. 2.(X, d) is complete and totally bounded (this is also equivalent to compactness for uniform spaces).[14] 3.(X, d) is sequentially compact; that is, every sequence in X has a convergent subsequence whose limit is in X (this is also equivalent to compactness for first-countable uniform spaces). 4.(X, d) is limit point compact (also called weakly countably compact); that is, every infinite subset of X has at least one limit point in X. 5.(X, d) is countably compact; that is, every countable open cover of X has a finite subcover. 6.(X, d) is an image of a continuous function from the Cantor set.[15] 7.Every decreasing nested sequence of nonempty closed subsets S1 ⊇ S2 ⊇ ... in (X, d) has a nonempty intersection. 8.Every increasing nested sequence of proper open subsets S1 ⊆ S2 ⊆ ... in (X, d) fails to cover X. (引用終り)
