| Title:
|
$\Cal P$-approximable compact spaces (English) |
| Author:
|
Tkačenko, Michael G. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
32 |
| Issue:
|
3 |
| Year:
|
1991 |
| Pages:
|
583-595 |
| . |
| Category:
|
math |
| . |
| Summary:
|
For every topological property $\Cal P$, we define the class of $\Cal P$-approximable spaces which consists of spaces X having a countable closed cover $\gamma $ such that the ``section'' $X(x,\gamma )= \bigcap \{F\in \gamma :x\in F\}$ has the property $\Cal P$ for each $x\in X$. It is shown that every $\Cal P$-approximable compact space has $\Cal P$, if $\Cal P$ is one of the following properties: countable tightness, $\aleph _0$-scatteredness with respect to character, $C$-closedness, sequentiality (the last holds under MA or $2^{\aleph _0}<2^{\aleph _1}$). Metrizable-approximable spaces are studied: every compact space in this class has a dense, Čech-complete, paracompact subspace; moreover, if $X$ is linearly ordered, then $X$ contains a dense metrizable subspace. (English) |
| Keyword:
|
$\Cal P$-approximable space |
| Keyword:
|
Lindelöf $\Sigma $-space |
| Keyword:
|
compact |
| Keyword:
|
metrizable |
| Keyword:
|
$C$-closed |
| Keyword:
|
sequential |
| Keyword:
|
linearly ordered |
| MSC:
|
54A20 |
| MSC:
|
54A25 |
| MSC:
|
54A35 |
| MSC:
|
54B05 |
| MSC:
|
54B10 |
| MSC:
|
54D20 |
| MSC:
|
54D30 |
| MSC:
|
54D55 |
| MSC:
|
54E35 |
| MSC:
|
54F05 |
| idZBL:
|
Zbl 0766.54019 |
| idMR:
|
MR1159804 |
| . |
| Date available:
|
2009-01-08T17:47:15Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118436 |
| . |
| Reference:
|
[1] Alexandroff P.S., Urysohn P.S.: Memoir on compact topological spaces (in Russian).Moscow, Nauka, 1971. MR 0370492 |
| Reference:
|
[2] Arhangel'skii A.V.: Bicompact sets and the topology of spaces.Trans. Mosc. Math. Soc. 13 (1965), 1-62. MR 0195046 |
| Reference:
|
[3] Arhangel'skii A.V.: On one class of spaces that contains all metrizable and all locally compact spaces (in Russian).Matem. Sb. 67 (1965), 55-88. MR 0190889 |
| Reference:
|
[4] Arhangel'skii A.V.: On bicompacta hereditarily satisfying the Souslin condition. Tightness and free sequences.Soviet. Math. Dokl. 12 (1971), 1253-1257. |
| Reference:
|
[5] Arhangel'skii A.V.: On compact spaces which are unions of certain collections of subspaces of special type.Comment. Math. Univ. Carolinae 17 (1976), 737-753. MR 0445453 |
| Reference:
|
[6] Arhangel'skii A.V.: On topologies weakly connected with orderings (in Russian).Dokl AN SSSR 238 (1978), 773-776. MR 0515022 |
| Reference:
|
[7] Arhangel'skii A.V.: On spaces of continuous functions with the pointwise convergence topology (in Russian).Dokl. AN SSSR 240 (1978), 505-508. MR 0500817 |
| Reference:
|
[8] Arhangel'skii A.V.: Structure and classification of topological spaces and cardinal invariants (in Russian).Uspechi Mat. Nauk 33 (1978), 29-84. MR 0526012 |
| Reference:
|
[9] Arhangel'skii A.V.: On invariants of type character and weight (in Russian).Trudy Mosc. Mat. Obsch. 38 (1979), 3-27. MR 0544935 |
| Reference:
|
[10] Balcar B., Pelant J., Simon P.: The space of ultrafilters on $\Bbb N$ covered by nowhere dense sets.Fund. Math. 110 (1980), 11-24. MR 0600576 |
| Reference:
|
[11] Engelking R.: General Topology.Warszawa, PWN, 1977. Zbl 0684.54001, MR 0500780 |
| Reference:
|
[12] Franklin S.P.: Spaces in which sequences suffice, II.Fund. Math. 57 (1967), 51-56. Zbl 0168.43502, MR 0222832 |
| Reference:
|
[13] Gul'ko S.P.: On the structure of spaces of continuous functions and their hereditary paracompactness (in Russian).Uspechi Mat. Nauk 34 (1979), 33-40. MR 0562814 |
| Reference:
|
[14] Ismail M.: Products of $C$-closed spaces.Houston J. Math. 10 (1984), 195-199. MR 0744903 |
| Reference:
|
[15] Ismail M., Nyikos P.: On spaces in which countably compact sets are closed, and hereditary properties.Topology Appl. 11 (1980), 281-292. Zbl 0434.54018, MR 0585273 |
| Reference:
|
[16] Juhász I.: Cardinal functions in topology.Math. Centre Tracts, 34, Amsterdam, 1971. MR 0340021 |
| Reference:
|
[17] Leiderman A.G.: On dense metrizable subspaces of Corson compact spaces (in Russian).Matem. Zametki 38 (1985), 440-449. MR 0811578 |
| Reference:
|
[18] Malyhin V.I.: On the tightness and the Souslin number of ${exp} X$ and of a product of spaces.Soviet Math. Dokl. 13 (1972), 496-499. |
| Reference:
|
[19] Malyhin V.I., Šapirovskii B.E.: Martin's axiom and properties of topological spaces (in Russian).Dokl. AN SSSR 213 (1973), 532-535. MR 0343241 |
| Reference:
|
[20] Nagami K.: $\Sigma $-spaces.Fund. Math. 65 (1969), 169-192. Zbl 0181.50701, MR 0257963 |
| Reference:
|
[21] Noble N.: Products with closed projections, II.Trans. Amer. Math. Soc. 160 (1971), 169-183. Zbl 0233.54004, MR 0283749 |
| Reference:
|
[22] Šapirovskii B.E.: On mappings onto Tychonoff cubes (in Russian).Uspechi Mat. Nauk 35 (1980), 122-130. MR 0580628 |
| Reference:
|
[23] Simon P.: Covering of a space by nowhere dense sets.Comment. Math. Univ. Carolinae 18 (1977), 755-761. Zbl 0373.54009, MR 0464133 |
| Reference:
|
[24] Sokolov G.A.: On some classes of compact spaces lying in $\Sigma $-products.Comment. Math. Univ. Carolinae 25 (1984), 219-231. Zbl 0577.54014, MR 0768809 |
| Reference:
|
[25] Štěpánek P., Vopěnka P.: Decomposition of metric spaces into nowhere dense sets.Comment. Math. Univ. Carolinae 8 (1967), 387-404, 567-568. MR 0225657 |
| Reference:
|
[26] Talagrand M.: Sur les espaces Banach faiblement $K$-analytiques.Comp. Rend. Acad. Sci., Ser. A 285 (1977), 119-122. MR 0440342 |
| Reference:
|
[27] Tkačenko M.G.: Some addition theorems in the class of compact spaces (in Russian).Sibir. Mat. J. 24 (1983), 135-143. MR 0731050 |
| Reference:
|
[28] Tkačenko M.G.: On compactness of countably compact spaces having additional structure.Trans. Mosc. Math. Soc. 1984, Issue 2, 149-167. |
| Reference:
|
[29] Tkačenko M.G.: The notion of o-tightness and $C$-embedded subspaces of products.Topology Appl. 15 (1983), 93-98. MR 0676970 |
| Reference:
|
[30] Todorčević S.: Cardinal functions on linearly ordered topological spaces, - Topology and order structures, Part I (Lubbock, Tex., 1980).p. 177-179, Math. Centre Tracts, 142, Math. Centrum, Amsterdam, 1981 (). MR 0630552 |
| Reference:
|
[31] Todorčević S.: Stationary sets and continuums.Publ. Inst. Math., Nouvelle sér. 27 (1981), 249-262. MR 0657114 |
| Reference:
|
[32] White H.E.: First-countable spaces that have special pseudobases.Canad. Math. Bull. 21 (1978), 103-112. MR 0482615 |
| . |
