pseudocompactness

Pseudocompactness is a topological property of a topological space. A topological space is pseudocompact if every real-valued continuous function on the space is bounded. Equivalently, a topological space is pseudocompact if every countable open cover has a finite subcover. This property is weaker than compactness. While every compact space is pseudocompact, the converse is not generally true.

For metric spaces, pseudocompactness is equivalent to compactness. However, in more general topological spaces, this equivalence does not hold. For example, the Tychonoff plank is a space that is pseudocompact but not compact. The concept is particularly relevant in the study of function spaces and in the context of the Hewitt-Marica theorem, which states that a Tychonoff space is pseudocompact if and only if it is the Rim proper.

Pseudocompactness is related to the notion of being "almost compact." A space is pseudocompact if and only if its Stone-Čech compactification is homeomorphic to the space itself. This means that there are no "missing points" in the space that would need to be added to make it compact. The study of pseudocompactness often involves constructing specific examples of spaces that exhibit this property but not compactness.