From countable compactness to absolute countable compactness

Authors:
Mary Ellen Rudin, Ian S. Stares and Jerry E. Vaughan

Journal:
Proc. Amer. Math. Soc. **125** (1997), 927-934

MSC (1991):
Primary 54D20; Secondary 54A35

DOI:
https://doi.org/10.1090/S0002-9939-97-04030-6

MathSciNet review:
1415367

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Abstract | References | Similar Articles | Additional Information

Abstract: We show that every countably compact space which is monotonically normal, almost 2-fully normal, radial , or with countable spread is absolutely countably compact. For the first two mentioned properties, we prove more general results not requiring countable compactness. We also prove that every monotonically normal, orthocompact space is finitely fully normal.

**1.**A. V. Arhangelskii,*On bicompacta hereditarily satisfying the Souslin condition. Tightness and free sequences.*Soviet Math. Dokl. 12 (1971), 1253-1257.**2.**A. V. Arhangelskii,*Structure and classfication of topological spaces*, Russian Math. Surveys 33 No. 5 (1978) 33-96.**3.**A. V. Arhangelskii,*Countably compact groups,*Math. Japonica 40 (1994) 39 - 53. MR**95i:54001****4.**Z. Balogh and M. E. Rudin,*Monotone normality*, Topology Appl. 47 (1992) 115-127. MR**94b:54065****5.**Angelo Bella,*Few remarks and questions on pseudo radial and related spaces*, to appear.**6.**A. Bella and Gerlitz,*On a condition for the pseudo radiality of a product*, Comment. Math. Univ. Carolin. (Prague) 33 (1992) 311-313. MR**93h:54001****7.**C. J. R. Borges,*A study of monotonically normal spaces*, Proc. Amer. Math. Soc. 38 (1973) 211-214. MR**48:2994****8.**E. K. van Douwen,*Simultaneous extension of continuous functions*, in Eric K. van Douwen, Collected Papers, Volume 1, J. van Mill, ed., North-Holland, Amsterdam, 1994. MR**96a:01047****9.**Ryszard Engelking, General Topology, Heldermann Verlag, Berlin 1989. MR**91c:54001****10.**Peter Fletcher and William F. Lindgren, Quasi-uniform Spaces, Marcel Dekker, New York 1982. MR**84h:54026****11.**R. W. Heath, D. J. Lutzer, and P. L. Zenor,*Monotonically normal spaces*, Trans. Amer. Math. Soc. 178 (1973) 481-493. MR**51:9030****12.**M. J. Mansfield,*Some generalizations of full normality*, Trans. Amer. Math. Soc. 86 (1957) 489-505. MR**20:273****13.**Michael V. Matveev,*Absolutely countably compact spaces*, Topology Appl. 58 (1994) 81-92. MR**95e:54035****14.**Jerry E. Vaughan,*On the product of a compact space with an absolutely countably compact space*, to appear in the Proceedings of the Vrije University Topology Conference, Amsterdam, 1994.

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Additional Information

**Mary Ellen Rudin**

Affiliation:
Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706

Email:
mrudin@math.wisc.edu

**Ian S. Stares**

Affiliation:
Department of Mathematical Sciences University of North Carolina at Greensboro Greensboro, North Carolina 27412

Email:
isstares@maths.ox.ac.uk

**Jerry E. Vaughan**

Email:
vaughanj@steffi.uncg.edu

DOI:
https://doi.org/10.1090/S0002-9939-97-04030-6

Keywords:
Countably compact,
absolutely countably compact,
monotonically normal,
property (a),
finitely fully normal,
almost 2-fully normal,
radial,
orthocompact,
countable spread

Received by editor(s):
September 10, 1995

Communicated by:
Franklin D. Tall

Article copyright:
© Copyright 1997
American Mathematical Society