Pseudocompact space

Properties related to pseudocompactness

• For a Tychonoff space X to be pseudocompact requires that every locally finite collection of non-empty open sets of X be finite. There are many equivalent conditions for pseudocompactness ; a large number of them are quoted in Stephenson 2003. Some historical remarks about earlier results can be found in Engelking 1989, p. 211.
• Every countably compact space is pseudocompact. For normal Hausdorff spaces the converse is true.
• As a consequence of the above result, every sequentially compact space is pseudocompact. The converse is true for metric spaces. As sequential compactness is an equivalent condition to compactness for metric spaces this implies that compactness is an equivalent condition to pseudocompactness for metric spaces also.
• The weaker result that every compact space is pseudocompact is easily proved: the image of a compact space under any continuous function is compact, and every compact set in a metric space is bounded.
• If Y is the continuous image of pseudocompact X, then Y is pseudocompact. Note that for continuous functions g : X → Y and h : Y → R, the composition of g and h, called f, is a continuous function from X to the real numbers. Therefore, f is bounded, and Y is pseudocompact.
• Let X be an infinite set given the particular point topology. Then X is neither compact, sequentially compact, countably compact, paracompact nor metacompact. However, since X is hyperconnected, it is pseudocompact. This shows that pseudocompactness doesn't imply any other form of compactness.
• For a Hausdorff space X to be compact requires that X be pseudocompact and realcompact.
• For a Tychonoff space X to be compact requires that X be pseudocompact and metacompact.

  Pseudocompact topological groups