*Definition [#u996b238]
-A topological space is said to be pseudocompact if its image under any continuous function to R is bounded.

*Property [#c3c52412]
-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>compact]] 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 the Heine–Borel theorem tells us that the compact subsets of R are precisely the closed and bounded subsets.
-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 (known) form of compactness.

*Reference [#mad4b919]
Lynn Arthur Steen and J. Arthur Seebach, Jr. ''Counterexamples in Topology'', Dover.