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

MathSciNet review:
1415367

Full-text PDF Free Access

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.**Alexander Arhangel′skii,*On countably compact and initially 𝜔₁-compact topological spaces and groups*, Math. Japon.**40**(1994), no. 1, 39–53. MR**1288016****4.**Z. Balogh and M. E. Rudin,*Monotone normality*, Topology Appl.**47**(1992), no. 2, 115–127. MR**1193194**, 10.1016/0166-8641(92)90066-9**5.**Angelo Bella,*Few remarks and questions on pseudo radial and related spaces*, to appear.**6.**A. Bella and J. Gerlits,*On a condition for the pseudo radiality of a product*, Comment. Math. Univ. Carolin.**33**(1992), no. 2, 311–313. MR**1189662****7.**Carlos R. Borges,*A study of monotonically normal spaces*, Proc. Amer. Math. Soc.**38**(1973), 211–214. MR**0324644**, 10.1090/S0002-9939-1973-0324644-4**8.**Eric K. van Douwen,*Collected papers. Vol. I, II*, North-Holland Publishing Co., Amsterdam, 1994. Edited and with a preface by Jan van Mill. MR**1281614****9.**Ryszard Engelking,*General topology*, 2nd ed., Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989. Translated from the Polish by the author. MR**1039321****10.**Peter Fletcher and William F. Lindgren,*Quasi-uniform spaces*, Lecture Notes in Pure and Applied Mathematics, vol. 77, Marcel Dekker, Inc., New York, 1982. MR**660063****11.**R. W. Heath, D. J. Lutzer, and P. L. Zenor,*Monotonically normal spaces*, Trans. Amer. Math. Soc.**178**(1973), 481–493. MR**0372826**, 10.1090/S0002-9947-1973-0372826-2**12.**M. J. Mansfield,*Some generalizations of full normality*, Trans. Amer. Math. Soc.**86**(1957), 489–505. MR**0093753**, 10.1090/S0002-9947-1957-0093753-5**13.**Michael V. Matveev,*Absolutely countably compact spaces*, Topology Appl.**58**(1994), no. 1, 81–92. MR**1280711**, 10.1016/0166-8641(94)90074-4**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.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
54D20,
54A35

Retrieve articles in all journals with MSC (1991): 54D20, 54A35

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