- Pseudocompact space
In

mathematics , in the field oftopology , atopological space is said to be**pseudocompact**if its image under anycontinuous function to**R**is bounded.**Conditions for pseudocompactness***Every countably compact space is pseudocompact. For

normal Hausdorff space s the converse is true.*As a consequence of the above result, every

sequentially compact space is pseudocompact. The converse is true formetric 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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2010.*

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