Pseudocompactness and closed subsets of products

Author:
James E. Joseph

Journal:
Proc. Amer. Math. Soc. **74** (1979), 338-342

MSC:
Primary 54D30; Secondary 54C30, 54C99

MathSciNet review:
524313

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 whenever and are continuous functions on a common domain into *Y* which agree on a dense subset of the domain.

**[BCM]**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**0097043**, 10.1090/S0002-9939-1958-0097043-2**[C]**W. W. Comfort,*A nonpseudocompact product space whose finite subproducts are pseudocompact*, Math. Ann.**170**(1967), 41–44. MR**0210070****[F]**Zdeněk Frolík,*The topological product of two pseudocompact spaces*, Czechoslovak Math. J**10(85)**(1960), 339–349 (English, with Russian summary). MR**0116304****[GJ]**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****[G1]**Irving Glicksberg,*Stone-Čech compactifications of products*, Trans. Amer. Math. Soc.**90**(1959), 369–382. MR**0105667**, 10.1090/S0002-9947-1959-0105667-4**[G2]**Irving Glicksberg,*The representation of functionals by integrals*, Duke Math. J.**19**(1952), 253–261. MR**0050168****[H]**Edwin Hewitt,*Rings of real-valued continuous functions. I*, Trans. Amer. Math. Soc.**64**(1948), 45–99. MR**0026239**, 10.1090/S0002-9947-1948-0026239-9**[J1]**J. E. Joseph,*Pseudocompactness via graphs and projections*(submitted).**[J2]**James E. Joseph,*Multifunctions and graphs*, Pacific J. Math.**79**(1978), no. 2, 509–529. MR**531332****[SS]**C. T. Scarborough and A. H. Stone,*Products of nearly compact spaces*, Trans. Amer. Math. Soc.**124**(1966), 131–147. MR**0203679**, 10.1090/S0002-9947-1966-0203679-7**[S]**R. M. Stephenson Jr.,*Pseudocompact spaces*, Trans. Amer. Math. Soc.**134**(1968), 437–448. MR**0232349**, 10.1090/S0002-9947-1968-0232349-6**[T]**Rodolfo Talamo,*Pseudocompact spaces and functionally determined uniformities*, Proc. Amer. Math. Soc.**56**(1976), 318–320. MR**0407808**, 10.1090/S0002-9939-1976-0407808-3

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
54D30,
54C30,
54C99

Retrieve articles in all journals with MSC: 54D30, 54C30, 54C99

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1979-0524313-7

Keywords:
Pseudocompactness,
filterbases,
graphs

Article copyright:
© Copyright 1979
American Mathematical Society