| Title:
|
On minimal strongly KC-spaces (English) |
| Author:
|
Sun, Weihua |
| Author:
|
Xu, Yuming |
| Author:
|
Li, Ning |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
59 |
| Issue:
|
2 |
| Year:
|
2009 |
| Pages:
|
305-316 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
${\rm KC}$-space |
| Keyword:
|
strongly ${\rm KC}$-space |
| Keyword:
|
${\rm FDS}$-property |
| Keyword:
|
maximal (countably) compact |
| MSC:
|
54A10 |
| MSC:
|
54D25 |
| MSC:
|
54D55 |
| idZBL:
|
Zbl 1224.54011 |
| idMR:
|
MR2532377 |
| . |
| Date available:
|
2010-07-20T15:09:26Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140482 |
| . |
| Reference:
|
[1] Alas, O. T., Tkachenko, M. G., Tkachuk, V. V., Wilson, R. G.: The ${\rm FDS}$-property and spaces in which compact sets are closed.Sci. Math. Jap. 61 (2005), 473-480. MR 2140109 |
| Reference:
|
[2] Alas, O. T., Wilson, R. G.: Spaces in which compact subsets are closed and the lattice of $\rm T_1$-topologies on a set.Commentat. Math. Univ. Carol. 43 (2002), 641-652. MR 2045786 |
| Reference:
|
[3] Cameron, D. E.: Maximal and minimal topologies.Trans. Amer. Math. Soc. 160 (1971), 229-248. Zbl 0202.22302, MR 0281142, 10.1090/S0002-9947-1971-0281142-7 |
| Reference:
|
[4] Engelking, R.: General Topology.PWN Warszawa (1977). Zbl 0373.54002, MR 0500780 |
| Reference:
|
[5] Fleissner, W. G.: A $T_B$-space which is not Katětov $T_B$.Rocky Mt. J. Math. 10 (1980), 661-663. Zbl 0448.54021, MR 0590229, 10.1216/RMJ-1980-10-3-661 |
| Reference:
|
[6] Kelley, J. L.: General Topology.Springer New York (1975). Zbl 0306.54002, MR 0370454 |
| Reference:
|
[7] Kunen, K., Vaughan, J. E.: Handbook of Set-Theoretic Topology.North Holland Amsterdam-New York-Oxford (1984). Zbl 0546.00022, MR 0776619 |
| Reference:
|
[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. MR 2126999 |
| Reference:
|
[9] Larson, R.: Complementary topological properties.Notices Am. Math. Soc. 20 (1973), 176. |
| Reference:
|
[10] Smythe, N., Wilkins, C. A.: Minimal Hausdorff and maximal compact spaces.J. Austr. Math. Soc. 3 (1963), 167-171. Zbl 0163.17201, MR 0154254, 10.1017/S1446788700027907 |
| Reference:
|
[11] Vidalis, T.: Minimal ${\rm KC}$-spaces are countably compact.Commentat. Math. Univ. Carol. 45 (2004), 543-547. MR 2103148 |
| Reference:
|
[12] Wilansky, A.: Between $\rm T_1$ and $\rm T_2$.Am. Math. Mon. 74 (1967), 261-266. MR 0208557 |
| . |
