- Pseudocompact space
mathematics, in the field of topology, a topological spaceis said to be pseudocompact if its image under any continuous functionto R is bounded.
Conditions for pseudocompactness
*Every countably compact space is pseudocompact. For
normal Hausdorff spaces the converse is true.
*As a consequence of the above result, every
sequentially compactspace 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 the
Heine-Borel theoremtells 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.
*springer|id=P/p075630|title=Pseudo-compact space|author=M.I. Voitsekhovskii
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