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.