Pseudocompact space

Symbol-free definition
A topological space is said to be pseudocompact if it satisfies the following equivalent properties:


 * 1) For any continuous map from the topological space to the real line, the image of the topological space is a closed and bounded subset of the real line.
 * 2) For any continuous map from the topological space to the real line, the image of the topological space is a bounded subset of the real line.
 * 3) Any continuous map from the topological space to the real line attains its absolute maximum and its absolute minimum. Note that this final formulation is equivalent to saying that the space satisfies the conclusion of the extreme value theorem, which is stated for compact spaces.

Related properties

 * Realcompact space