# pseudocompact space

A topological space $X$ is said to be pseudocompact if every continuous function $f\colon X\to\mathbb{R}$ has bounded image.

All countably compact spaces (which includes all compact spaces and all sequentially compact spaces) are pseudocompact. A metric space is pseudocompact if and only if it is compact. A Hausdorff normal space is pseudocompact if and only if it is countably compact.

 Title pseudocompact space Canonical name PseudocompactSpace Date of creation 2013-03-22 14:20:36 Last modified on 2013-03-22 14:20:36 Owner yark (2760) Last modified by yark (2760) Numerical id 7 Author yark (2760) Entry type Definition Classification msc 54D30 Synonym pseudo compact space Synonym pseudo-compact space Related topic LimitPointCompact Defines pseudocompact Defines pseudocompactness Defines pseudo-compact Defines pseudo-compactness Defines pseudo compact Defines pseudo compactness