pseudocompact space

A topological space $X$ is said to be pseudocompact if every continuous function $f:X\to ℝ$ 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.