On minimal strongly KC-spaces

Weihua Sun; Yuming Xu; Ning Li

Czechoslovak Mathematical Journal (2009)

  • Volume: 59, Issue: 2, page 305-316
  • ISSN: 0011-4642

Abstract

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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 Katětov strongly KC -spaces and finally, we generalize one result of Alas and Wilson on Katětov- KC spaces.

How to cite

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Sun, Weihua, Xu, Yuming, and Li, Ning. "On minimal strongly KC-spaces." Czechoslovak Mathematical Journal 59.2 (2009): 305-316. <http://eudml.org/doc/37925>.

@article{Sun2009,
abstract = {In this article we introduce the notion of strongly $\{\rm KC\}$-spaces, that is, those spaces in which countably compact subsets are closed. We find they have good properties. We prove that a space $(X, \tau )$ is maximal countably compact if and only if it is minimal strongly $\{\rm KC\}$, and apply this result to study some properties of minimal strongly $\{\rm KC\}$-spaces, some of which are not possessed by minimal $\{\rm KC\}$-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 $\{\rm KC\}$-space of cardinality less than $c$ has the $\{\rm FDS \}$-property. Using this we obtain a characterization of Katětov strongly $\{\rm KC\}$-spaces and finally, we generalize one result of Alas and Wilson on Katětov-$\{\rm KC\}$ spaces.},
author = {Sun, Weihua, Xu, Yuming, Li, Ning},
journal = {Czechoslovak Mathematical Journal},
keywords = {$\{\rm KC\}$-space; strongly $\{\rm KC\}$-space; $\{\rm FDS\}$-property; maximal (countably) compact; -space, strongly -space; -property; maximal (countably) compact},
language = {eng},
number = {2},
pages = {305-316},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On minimal strongly KC-spaces},
url = {http://eudml.org/doc/37925},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Sun, Weihua
AU - Xu, Yuming
AU - Li, Ning
TI - On minimal strongly KC-spaces
JO - Czechoslovak Mathematical Journal
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 59
IS - 2
SP - 305
EP - 316
AB - In this article we introduce the notion of strongly ${\rm KC}$-spaces, that is, those spaces in which countably compact subsets are closed. We find they have good properties. We prove that a space $(X, \tau )$ is maximal countably compact if and only if it is minimal strongly ${\rm KC}$, and apply this result to study some properties of minimal strongly ${\rm KC}$-spaces, some of which are not possessed by minimal ${\rm KC}$-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 ${\rm KC}$-space of cardinality less than $c$ has the ${\rm FDS }$-property. Using this we obtain a characterization of Katětov strongly ${\rm KC}$-spaces and finally, we generalize one result of Alas and Wilson on Katětov-${\rm KC}$ spaces.
LA - eng
KW - ${\rm KC}$-space; strongly ${\rm KC}$-space; ${\rm FDS}$-property; maximal (countably) compact; -space, strongly -space; -property; maximal (countably) compact
UR - http://eudml.org/doc/37925
ER -

References

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  2. Alas, O. T., Wilson, R. G., Spaces in which compact subsets are closed and the lattice of T 1 -topologies on a set, Commentat. Math. Univ. Carol. 43 (2002), 641-652. (2002) MR2045786
  3. Cameron, D. E., 10.1090/S0002-9947-1971-0281142-7, Trans. Amer. Math. Soc. 160 (1971), 229-248. (1971) Zbl0202.22302MR0281142DOI10.1090/S0002-9947-1971-0281142-7
  4. Engelking, R., General Topology, PWN Warszawa (1977). (1977) Zbl0373.54002MR0500780
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  6. Kelley, J. L., General Topology, Springer New York (1975). (1975) Zbl0306.54002MR0370454
  7. Kunen, K., Vaughan, J. E., Handbook of Set-Theoretic Topology, North Holland Amsterdam-New York-Oxford (1984). (1984) Zbl0546.00022MR0776619
  8. Kunzi, H.-P. A., Zypen, D. van der, Maximal (sequentially) compact topologies, In: Proc. North-West Eur. categ. sem., Berlin, Germany, March 28-29, 2003 World Scientific River Edge (2004), 173-187. (2004) MR2126999
  9. Larson, R., Complementary topological properties, Notices Am. Math. Soc. 20 (1973), 176. (1973) 
  10. Smythe, N., Wilkins, C. A., 10.1017/S1446788700027907, J. Austr. Math. Soc. 3 (1963), 167-171. (1963) Zbl0163.17201MR0154254DOI10.1017/S1446788700027907
  11. Vidalis, T., Minimal KC -spaces are countably compact, Commentat. Math. Univ. Carol. 45 (2004), 543-547. (2004) MR2103148
  12. Wilansky, A., Between T 1 and T 2 , Am. Math. Mon. 74 (1967), 261-266. (1967) MR0208557

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