Pseudocompactness and closed subsets of products

Joseph, James E.

doi:10.1090/S0002-9939-1979-0524313-7

Pseudocompactness and closed subsets of products
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Proc. Amer. Math. Soc. 74 (1979), 338-342 Request permission

Abstract:

This paper contains several new characterizations of arbitrary pseudocompact spaces, i.e. spaces characterized by the property that all continuous real-valued functions on the space are bounded. These characterizations parallel known characterizations of Hausdorff spaces including the useful and well-known result that a space Y is Hausdorff if and only if $\phi = \alpha$ whenever $\phi$ and $\alpha$ are continuous functions on a common domain into Y which agree on a dense subset of the domain.

References

R. W. Bagley, E. H. Connell, and J. D. McKnight Jr., On properties characterizing pseudo-compact spaces, Proc. Amer. Math. Soc. 9 (1958), 500–506. MR 97043, DOI 10.1090/S0002-9939-1958-0097043-2
W. W. Comfort, A nonpseudocompact product space whose finite subproducts are pseudocompact, Math. Ann. 170 (1967), 41–44. MR 210070, DOI 10.1007/BF01362285
Zdeněk Frolík, The topological product of two pseudocompact spaces, Czechoslovak Math. J. 10(85) (1960), 339–349 (English, with Russian summary). MR 116304
Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0116199
Irving Glicksberg, Stone-Čech compactifications of products, Trans. Amer. Math. Soc. 90 (1959), 369–382. MR 105667, DOI 10.1090/S0002-9947-1959-0105667-4
Irving Glicksberg, The representation of functionals by integrals, Duke Math. J. 19 (1952), 253–261. MR 50168
Edwin Hewitt, Rings of real-valued continuous functions. I, Trans. Amer. Math. Soc. 64 (1948), 45–99. MR 26239, DOI 10.1090/S0002-9947-1948-0026239-9

Pseudocompactness via graphs and projections

James E. Joseph, Multifunctions and graphs, Pacific J. Math. 79 (1978), no. 2, 509–529. MR 531332
C. T. Scarborough and A. H. Stone, Products of nearly compact spaces, Trans. Amer. Math. Soc. 124 (1966), 131–147. MR 203679, DOI 10.1090/S0002-9947-1966-0203679-7
R. M. Stephenson Jr., Pseudocompact spaces, Trans. Amer. Math. Soc. 134 (1968), 437–448. MR 232349, DOI 10.1090/S0002-9947-1968-0232349-6
Rodolfo Talamo, Pseudocompact spaces and functionally determined uniformities, Proc. Amer. Math. Soc. 56 (1976), 318–320. MR 407808, DOI 10.1090/S0002-9939-1976-0407808-3