A countably compact space and its products

Author:
J. E. Vaughan

Journal:
Proc. Amer. Math. Soc. **71** (1978), 133-137

MSC:
Primary 54D30

DOI:
https://doi.org/10.1090/S0002-9939-1978-0474206-8

MathSciNet review:
0474206

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is known that if a topological space *X* is totally countably compact, then (a) (the product of copies of *X*) is countably compact, and (b) for every countably compact space *Y*, the product is countably compact. The main result of this paper is the construction of a space which satisfies (a) and (b), and is not totally countably compact. The example is , where is the type of a certain kind of ultrafilter on the natural numbers .

**[1]**A. R. Bernstein,*A new kind of compactness for topological spaces*, Fund. Math.**66**(1970), 185-193. MR**0251697 (40:4924)****[2]**W. W. Comfort and S. Negrepontis,*The theory of ultrafilters*, Die Grundlehren der math. Wissenschaften, Bd. 211, Springer-Verlag, New York, 1974. MR**0396267 (53:135)****[3]**Z. Frolík,*The topological product of countably compact spaces*, Czechoslovak Math. J.**10**(**85**) (1960), 329-338. MR**0117705 (22:8480)****[4]**-,*The topological product of two pseudocompact spaces*, Czechoslovak Math. J.**10**(**85**) 1960, 339-349. MR**0116304 (22:7099)****[5]**John Ginsburg and Victor Saks,*Some applications of ultrafilters in topology*, Pacific J. Math.**57**(1975), 403-418. MR**0380736 (52:1633)****[6]**T. Isiwata,*Some classes of countably compact spaces*, Czechoslovak Math. J.**14**(**89**) (1964), 22-26. MR**0165484 (29:2766)****[7]**K. Kunen,*Some points in*, Proc. Cambridge Philos. Soc.**80**(1976), 385-398. MR**0427070 (55:106)****[8]**-,*Another point in*(preprint).**[9]**Victor Saks,*Ultrafilter invariants in topological spaces*, Trans. Amer. Math. Soc. (to appear). MR**492291 (80h:54003)****[10]**J. E. Vaughan,*Products of topological spaces*, General Topology and Appl. (to appear). MR**0493935 (58:12891)**

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

Retrieve articles in all journals with MSC: 54D30

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1978-0474206-8

Keywords:
Countably compact,
pseudocompact,
*r*-compactness,
totally countably compact,
Rudin-Keisler order,
types,
*P*-points

Article copyright:
© Copyright 1978
American Mathematical Society