Boundedly Compact Space

A metric space X is boundedly compact if all closed bounded subsets of X are compact. Every boundedly compact metric space is complete. (This is a generalization of the Bolzano-Weierstrass theorem.)

Every complete Riemannian manifold is boundedly compact. This is part of or a consequence of the Hopf-Rinow theorem.

See also

Bolzano-Weierstrass Theorem, Bounded, Complete Metric Space, Hopf-Rinow Theorem, Metric Space

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Cite this as:

Weisstein, Eric W. "Boundedly Compact Space." From MathWorld--A Wolfram Web Resource.

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