| Title:
|
An example of a space whose all continuous mappings are almost injective (English) |
| Author:
|
Iturralde, Pablo Mendoza |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
42 |
| Issue:
|
3 |
| Year:
|
2001 |
| Pages:
|
535-544 |
| . |
| Category:
|
math |
| . |
| Summary:
|
We show that all continuous maps of a space $X$ onto second countable spaces are pseudo-open if and only if every discrete family of nonempty $G_\delta $-subsets of $X$ is finite. We also prove under CH that there exists a dense subspace $X$ of the real line $\Bbb R$, such that every continuous map of $X$ is almost injective and $X$ cannot be represented as $K\cup Y$, where $K$ is compact and $Y$ is countable. This partially answers a question of V.V. Tkachuk in [Tk]. We show that for a compact $X$, all continuous maps of $X$ onto second countable spaces are almost injective if and only if it is scattered. We give an example of a non-compact space $Z$ such that every continuous map of $Z$ onto a second countable space is almost injective but $Z$ is not scattered. (English) |
| Keyword:
|
almost compact map |
| Keyword:
|
pseudo-open map |
| Keyword:
|
almost injective map |
| Keyword:
|
discrete family |
| Keyword:
|
scattered |
| MSC:
|
54C10 |
| MSC:
|
54D18 |
| MSC:
|
54D20 |
| MSC:
|
54D30 |
| MSC:
|
54E52 |
| idZBL:
|
Zbl 1053.54022 |
| idMR:
|
MR1860242 |
| . |
| Date available:
|
2009-01-08T19:15:53Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119268 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
[Mr] Mrówka S.: Some set-theoric constructions in topology.Fund. Math. 94 (1977), 83-92. MR 0433388 |
| Reference:
|
[Ok] Okromeshko N.G.: Properties of quotient mappings and representation as continuous images (in Russian).Candidates Dissertation, Moscow, 1982. |
| Reference:
|
[Pa] Pasynkov B.A.: On open mappings.Doklady Akad. Nauk SSSR 175 (1967), 292-295; English transl.: Soviet Math. Dokl. 8 (1967). Zbl 0155.50203, MR 0217773 |
| Reference:
|
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| Reference:
|
[Tk] Tkachuk V.V.: Spaces that are projective with respect to classes of mappings.Trans. Moscow Math. Soc. 50 (1988), 139-156. Zbl 0662.54007, MR 0912056 |
| Reference:
|
[Tk1] Tkachuk V.V.: Remainders over discretes-some applications.Vestnik Moskov. Univ. Matematika 45 (1990), 4 18-21. MR 1086601 |
| Reference:
|
[Tka] Tkačenko M.G.: The Souslin property in free topological groups over compact spaces.Math. Notes 34 (1983), 790-793. MR 0722229 |
| Reference:
|
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| . |
