Abstract
In this article we introduce the notion of strongly KC-spaces, that is, those spaces in which countably compact subsets are closed. We find they have good properties. We prove that a space (X, τ) is maximal countably compact if and only if it is minimal strongly KC, and apply this result to study some properties of minimal strongly KC-spaces, some of which are not possessed by minimal KC-spaces. We also give a positive answer to a question proposed by O.T. Alas and R.G. Wilson, who asked whether every countably compact KC-space of cardinality less than c has the FDS-property. Using this we obtain a characterization of Katítov strongly KC-spaces and finally, we generalize one result of Alas and Wilson on Katìtov-KC spaces.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
O. T. Alas, M. G. Tkachenko, V. V. Tkachuk, R. G. Wilson: The FDS-property and spaces in which compact sets are closed. Sci. Math. Jap. 61 (2005), 473–480.
O. T. Alas, R. G. Wilson: Spaces in which compact subsets are closed and the lattice of T1-topologies on a set. Commentat. Math. Univ. Carol. 43 (2002), 641–652.
D. E. Cameron: Maximal and minimal topologies. Trans. Amer. Math. Soc. 160 (1971), 229–248.
R. Engelking: General Topology. PWN, Warszawa, 1977.
W. G. Fleissner: A T B -space which is not Katětov T B . Rocky Mt. J. Math. 10 (1980), 661–663.
J. L. Kelley: General Topology. Springer, New York, 1975.
K. Kunen, J. E. Vaughan: Handbook of Set-Theoretic Topology. North Holland, Amsterdam-New York-Oxford, 1984.
H.-P. A. Kunzi, D. van der Zypen: Maximal (sequentially) compact topologies. In: Proc. North-West Eur. categ. sem., Berlin, Germany, March 28–29, 2003. World Scientific, River Edge, 2004, pp. 173–187.
R. Larson: Complementary topological properties. Notices Am. Math. Soc. 20 (1973), 176.
N. Smythe, C. A. Wilkins: Minimal Hausdorff and maximal compact spaces. J. Austr. Math. Soc. 3 (1963), 167–171.
T. Vidalis: Minimal KC-spaces are countably compact. Commentat. Math. Univ. Carol. 45 (2004), 543–547.
A. Wilansky: Between T1 and T2. Am. Math. Mon. 74 (1967), 261–266.
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported by NSFC of China (No. 10671173).
Rights and permissions
About this article
Cite this article
Sun, W., Xu, Y. & Li, N. On minimal strongly KC-spaces. Czech Math J 59, 305–316 (2009). https://doi.org/10.1007/s10587-009-0022-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10587-009-0022-6