| Title:
|
A construction of a Fréchet-Urysohn space, and some convergence concepts (English) |
| Author:
|
Arhangel'skii, A. V. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
51 |
| Issue:
|
1 |
| Year:
|
2010 |
| Pages:
|
99-112 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Some strong versions of the Fréchet-Urysohn property are introduced and studied. We also strengthen the concept of countable tightness and generalize the notions of first-countability and countable base. A construction of a topological space is described which results, in particular, in a Tychonoff countable Fréchet-Urysohn space which is not first-countable at any point. It is shown that this space can be represented as the image of a countable metrizable space under a continuous pseudoopen mapping. On the other hand, if a topological group $G$ is an image of a separable metrizable space under a pseudoopen continuous mapping, then $G$ is metrizable (Theorem 5.6). Several other applications of the techniques developed below to the study of pseudoopen mappings and intersections of topologies are given (see Theorem 5.17). (English) |
| Keyword:
|
first-countable |
| Keyword:
|
Fréchet-Urysohn |
| Keyword:
|
countably compact |
| Keyword:
|
closure-sensor |
| Keyword:
|
topological group |
| Keyword:
|
strong FU-sensor |
| Keyword:
|
pseudoopen mapping |
| Keyword:
|
side-base |
| Keyword:
|
$\omega $-Fréchet-Urysohn space |
| MSC:
|
54D20 |
| MSC:
|
54G20 |
| MSC:
|
54J99 |
| idZBL:
|
Zbl 1224.54055 |
| idMR:
|
MR2666083 |
| . |
| Date available:
|
2010-05-21T12:36:10Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140083 |
| . |
| Reference:
|
[1] Arhangel'skii A.V.: Structure and classification of topological spaces and cardinal invariants.Russian Math. Surveys 33 (1978), 33–96. MR 0526012, 10.1070/RM1978v033n06ABEH003884 |
| Reference:
|
[2] Arhangel'skii A.V.: Hurewicz spaces, analytic sets, and fan-tightness of function spaces.Dokl. Akad. Nauk SSSR 287:3 (1986), 525–528; English translation: Soviet Math. Dokl. 33:2 (1986), 396–399. MR 0837289 |
| Reference:
|
[3] Arhangel'skii A.V., Bella A.: Countable fan-tightness versus countable tightness.Comment. Math. Univ. Carolin. 37:3 (1996), 565–576. MR 1426921 |
| Reference:
|
[4] Arhangel'skii A.V. Ponomarev V.I.: Fundamentals of General Topology in Problems and Exercises.Izdat. “Nauka”, Moscow, 1974, 423 pp. (in Russian); English translation: ser. Mathematics and its Applications, D. Reidel Publishing Co., Dordrecht-Boston, Mass., 1984. xvi+415 pp.; Polish translation: Panstwowe Wydawnictwo Naukowe (PWN), Warsaw, 1986. 456 pp. MR 0785749 |
| Reference:
|
[5] Arhangel'skii A.V. Tkachenko M.G.: Topological Groups and Related Structures.Atlantis Press, Amsterdam-Paris, 2008. MR 2433295 |
| Reference:
|
[6] Engelking R.: General Topology.Sigma Series in Pure Mathematics, 6, Heldermann, Berlin, revised ed., 1989. Zbl 0684.54001, MR 1039321 |
| Reference:
|
[7] Michael E.A.: A quintuple quotient quest.General Topology Appl. 2 (1972), 91–138. Zbl 0238.54009, MR 0309045, 10.1016/0016-660X(72)90040-2 |
| Reference:
|
[8] Nyikos P.J.: Subsets of $\omega ^{\omega }$ and the Fréchet-Urysohn and $\alpha _i$-properties.Topology Appl. 48 (1992), 91–116. MR 1195504, 10.1016/0166-8641(92)90021-Q |
| . |
