| Title:
|
When is $\bold N$ Lindelöf? (English) |
| Author:
|
Herrlich, Horst |
| Author:
|
Strecker, George E. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
38 |
| Issue:
|
3 |
| Year:
|
1997 |
| Pages:
|
553-556 |
| . |
| Category:
|
math |
| . |
| Summary:
|
Theorem. In ZF (i.e., Zermelo-Fraenkel set theory without the axiom of choice) the following conditions are equivalent: (1) $\Bbb N$ is a Lindelöf space, (2) $\Bbb Q$ is a Lindelöf space, (3) $\Bbb R$ is a Lindelöf space, (4) every topological space with a countable base is a Lindelöf space, (5) every subspace of $\Bbb R$ is separable, (6) in $\Bbb R$, a point $x$ is in the closure of a set $A$ iff there exists a sequence in $A$ that converges to $x$, (7) a function $f:\Bbb R\rightarrow \Bbb R$ is continuous at a point $x$ iff $f$ is sequentially continuous at $x$, (8) in $\Bbb R$, every unbounded set contains a countable, unbounded set, (9) the axiom of countable choice holds for subsets of $\Bbb R$. (English) |
| Keyword:
|
axiom of choice |
| Keyword:
|
axiom of countable choice |
| Keyword:
|
Lindelöf space |
| Keyword:
|
separable space |
| Keyword:
|
(sequential) continuity |
| Keyword:
|
(Dedekind-) finiteness |
| MSC:
|
03E25 |
| MSC:
|
04A25 |
| MSC:
|
26A03 |
| MSC:
|
26A15 |
| MSC:
|
54A35 |
| MSC:
|
54D20 |
| idZBL:
|
Zbl 0938.54008 |
| idMR:
|
MR1485075 |
| . |
| Date available:
|
2009-01-08T18:36:09Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118952 |
| . |
| Reference:
|
Bentley H.L., Herrlich H.: Countable choice and pseudometric spaces.in preparation. Zbl 0922.03068 |
| Reference:
|
Herrlich H.: Compactness and the Axiom of Choice.Appl. Categ. Struct. 4 (1996), 1-14. Zbl 0881.54027, MR 1393958 |
| Reference:
|
Herrlich H., Steprāns J.: Maximal Filters, continuity, and choice principles.to appear in Quaestiones Math. MR 1625478 |
| Reference:
|
Jaegermann M.: The axiom of choice and two definitions of continuity.Bulletin de l'Acad. Polonaise des Sciences, Ser. Math. 13 (1965), 699-704. Zbl 0252.02059, MR 0195711 |
| Reference:
|
Jech T.: Eine Bemerkung zum Auswahlaxiom.Časopis pro pěstování matematiky 9 (1968), 30-31. Zbl 0167.27402, MR 0233706 |
| Reference:
|
Sierpiński W.: Sur le rôle de l'axiome de M. Zermelo dans l'Analyse moderne.Compt. Rendus Hebdomadaires des Séances de l'Academie des Sciences, Paris 193 (1916), 688-691. |
| Reference:
|
Sierpiński W.: L'axiome de M. Zermelo et son rôle dans la théorie des ensembles et l'analyse.Bulletin de l'Académie des Sciences de Cracovie, Classe des Sciences Math., Sér. A (1918), 97-152. |
| . |
