Pseudocompact space


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.

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 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 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.

References

*springer|id=P/p075630|title=Pseudo-compact space|author=M.I. Voitsekhovskii
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