For aTychonoff spaceX to bepseudocompact requires that everylocally finite collection ofnon-emptyopen sets ofX befinite. There are many equivalent conditions for pseudocompactness (sometimes someseparation axiom should be assumed); a large number of them are quoted in Stephenson 2003. Some historical remarks about earlier results can be found in Engelking 1989, p. 211.
As a consequence of the above result, everysequentially compact space is pseudocompact. The converse is true formetric spaces. As sequential compactness is an equivalent condition tocompactness 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.
IfY is the continuous image of pseudocompactX, thenY is pseudocompact. Note that for continuous functionsg : X → Y andh : Y → R, thecomposition ofg andh, calledf, is a continuous function fromX to the real numbers. Therefore,f is bounded, andY is pseudocompact.
LetX be an infinite set given theparticular point topology. ThenX is neither compact, sequentially compact, countably compact,paracompact normetacompact (although it isorthocompact). However, sinceX ishyperconnected, it is pseudocompact. This shows that pseudocompactness doesn't imply any of these other forms of compactness.
For aHausdorff spaceX to becompact requires thatX bepseudocompact andrealcompact (see Engelking 1968, p. 153).
For aTychonoff spaceX to becompact requires thatX bepseudocompact andmetacompact (see Watson).
A relatively refined theory is available for pseudocompacttopological groups.[2] In particular,W. W. Comfort andKenneth A. Ross proved that a product of pseudocompact topological groups is still pseudocompact (this might fail for arbitrary topological spaces).[3]
^See, for example,Mikhail Tkachenko, Topological Groups: Between Compactness and-boundedness, inMirek Husek andJan van Mill (eds.), Recent Progress in General Topology II, 2002 Elsevier Science B.V.
^Comfort, W. W. and Ross, K. A., Pseudocompactness and uniform continuity in topological groups, Pacific J. Math. 16, 483-496, 1966.[2]
Stephenson, R.M. Jr (2003),Pseudocompact Spaces, Chapter d-7 in Encyclopedia of General Topology, Edited by: Klaas Pieter Hart, Jun-iti Nagata and Jerry E. Vaughan, Pages 177-181, Amsterdam: Elsevier B. V..
