Pseudocompact space

In mathematics, in the field of topology, a topological space is said to be pseudocompact if its image under any continuous function to R is bounded. Many authors include the request that the space is completely regular right in the definition of pseudocompactness. Pseudocompact spaces were defined by Edwin Hewitt in 1948.

A relatively refined theory is available for pseudocompact topological groups. In particular, W. W. Comfort and Kenneth A. Ross proved that a product of pseudocompact topological groups is still pseudocompact (this might fail for arbitrary topological spaces).

...
Wikipedia