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.

Properties related to pseudocompactness

• In order that a Tychonoff space X be pseudocompact it is necessary and sufficient that every locally finite collection of non-empty open sets of X be finite. A series of equivalent conditions was given by Kerstan and Yan-Min and other authors (see the references).

• 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.

• In order that a Hausdorff space X be compact it is necessary and sufficient that X be pseudocompact and realcompact (see Engelking, p. 153).

• In order that a Tychonoff space X be compact it is necessary and sufficient that X be pseudocompact and metacompact (see Watson).

See also

• Compact space
• Paracompact space
• Normal space
• Realcompact space
• Metacompact space
• Tychonoff space

References

• Kerstan, Johannes (1957), "Zur Charakterisierung der pseudokompakten Räume", Mathematische Nachrichten 16 (5–6): 289–293, doi:10.1002/mana.19570160505 .
• W. Stephen, Watson (1981), "Pseudocompact metacompact spaces are compact", Proc. Amer. Math. Soc. 81: 151–152, doi:10.1090/s0002-9939-1981-0589159-1 .
• Yan-Min, Wang (1988), "New characterisations of pseudocompact spaces", Bull. Austral. Math. Soc. 38 (2): 293–298, doi:10.1017/S0004972700027568 .
• Engelking, Ryszard (1968), Outline of General Topology, translated from Polish, Amsterdam: North-Holland .
• Willard, Stephen (1970), General Topology, Reading, Mass.: Addison-Wesley .
• M.I. Voitsekhovskii (2001), "Pseudo-compact space", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 .
• Pseudocompact space at PlanetMath.org. .