| Title:
|
Products of Lindelöf $T_2$-spaces are Lindelöf – in some models of ZF (English) |
| Author:
|
Herrlich, Horst |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
43 |
| Issue:
|
2 |
| Year:
|
2002 |
| Pages:
|
319-333 |
| . |
| Category:
|
math |
| . |
| Summary:
|
The stability of the Lindelöf property under the formation of products and of sums is investigated in ZF (= Zermelo-Fraenkel set theory without AC, the axiom of choice). It is • not surprising that countable summability of the Lindelöf property requires some weak choice principle, • highly surprising, however, that productivity of the Lindelöf property is guaranteed by a drastic failure of AC, • amusing that finite summability of the Lindelöf property takes place if either some weak choice principle holds or if AC fails drastically. Main results: 1. Lindelöf = compact for $T_1$-spaces iff $\text{\bf CC}(\Bbb R)$, the axiom of countable choice for subsets of the reals, fails. 2. Lindelöf $T_1$-spaces are finitely productive iff $\text{\bf CC}(\Bbb R)$ fails. 3. Lindelöf $T_2$-spaces are productive iff $\text{\bf CC}(\Bbb R)$ fails and $\text{\bf BPI}$, the Boolean prime ideal theorem, holds. 4. Arbitrary products and countable sums of compact $T_1$-spaces are Lindelöf iff $\text{\bf AC}$ holds. 5. Lindelöf spaces are countably summable iff $\text{\bf CC}$, the axiom of countable choice, holds. 6. Lindelöf spaces are finitely summable iff either $\text{\bf CC}$ holds or $\text{\bf CC}(\Bbb R)$ fails. 7. Lindelöf $T_2$-spaces are $T_3$ spaces iff $\text{\bf CC}(\Bbb R)$ fails. 8. Totally disconnected Lindelöf $T_2$-spaces are zerodimensional iff $\text{\bf CC}(\Bbb R)$ fails. (English) |
| Keyword:
|
axiom of choice |
| Keyword:
|
axiom of countable choice |
| Keyword:
|
Lindelöf space |
| Keyword:
|
compact space |
| Keyword:
|
product |
| Keyword:
|
sum |
| MSC:
|
03E25 |
| MSC:
|
54A35 |
| MSC:
|
54B10 |
| MSC:
|
54D20 |
| MSC:
|
54D30 |
| idZBL:
|
Zbl 1072.03029 |
| idMR:
|
MR1922130 |
| . |
| Date available:
|
2009-01-08T19:22:19Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119322 |
| . |
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