| Title:
|
$F_\sigma $-absorbing sequences in hyperspaces of subcontinua (English) |
| Author:
|
Gladdines, Helma |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
34 |
| Issue:
|
4 |
| Year:
|
1993 |
| Pages:
|
729-745 |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $\Cal D$ denote a true dimension function, i.e., a dimension function such that $\Cal D(\Bbb R^n) = n$ for all $n$. For a space $X$, we denote the hyperspace consisting of all compact connected, non-empty subsets by $C(X)$. If $X$ is a countable infinite product of non-degenerate Peano continua, then the sequence $(\Cal D_{\geq n}(C(X)))_{n=2}^\infty$ is $F_\sigma$-absorbing in $C(X)$. As a consequence, there is a homeomorphism $h: C(X)\rightarrow Q^\infty$ such that for all $n$, $h[\{A \in C(X) : \Cal D(A) \geq n+1\}] = B^n \times Q \times Q \times \dots $, where $B$ denotes the pseudo boundary of the Hilbert cube $Q$. It follows that if $X$ is a countable infinite product of non-degenerate Peano continua then $\Cal D_{\geq n}(C(X))$ is an $F_\sigma$-absorber (capset) for $C(X)$, for every $n \geq 2$. Let $\operatorname{dim}$ denote covering dimension. It is known that there is an example of an everywhere infinite dimensional Peano continuum $X$ that contains arbitrary large $n$-cubes, such that for every $k \in \Bbb N$, the sequence $(\operatorname{dim}_{\geq n}(C(X^k)))_{n=2}^\infty$ is not $F_\sigma$-absorbing in $C(X^k)$. So our result is in some sense the best possible. (English) |
| Keyword:
|
Hilbert cube |
| Keyword:
|
absorbing system |
| Keyword:
|
$F_\sigma$ |
| Keyword:
|
$F_{\sigma \delta}$ |
| Keyword:
|
capset |
| Keyword:
|
Peano continuum |
| Keyword:
|
hyperspace |
| Keyword:
|
hyperspace of subcontinua |
| Keyword:
|
covering dimension |
| Keyword:
|
cohomological dimension |
| MSC:
|
54B20 |
| MSC:
|
54F15 |
| MSC:
|
54F45 |
| MSC:
|
55M10 |
| MSC:
|
57N20 |
| idZBL:
|
Zbl 0813.57020 |
| idMR:
|
MR1263802 |
| . |
| Date available:
|
2009-01-08T18:07:55Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118630 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
[3] Curtis D.W.: Boundary sets in the Hilbert cube.Top. Appl. 20 (1985), 201-221. Zbl 0575.57008, MR 0804034 |
| Reference:
|
[4] Curtis D.W., Nhu N.T.: Hyperspaces of finite subsets which are homeomorphic to $\aleph _0$-dimensional linear metric space.Top. Appl. 19 (1985), 251-260. MR 0794488 |
| Reference:
|
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| Reference:
|
[6] Gladdines H., van Mill J.: Hyperspaces of infinite-dimensional compacta.Comp. Math. 88 (1993), 143-153. Zbl 0830.57013, MR 1237918 |
| Reference:
|
[7] Gladdines H.: $F_\sigma $-absorbing sequences in hyperspaces of compact sets.Bull. Pol. Ac. Sci. vol 40 (3) (1992). MR 1401869 |
| Reference:
|
[8] Gladdines H., Baars J., van Mill J.: Absorbing systems in infinite-dimensional manifolds.Topology Appl. 50 (1993), 147-182. Zbl 0794.57005, MR 1217483 |
| Reference:
|
[9] Dobrowolski T., Rubin L.R.: The hyperspace of infinite-dimensional compacta for covering and cohomological dimension are homeomorphic.preprint, to appear. MR 1267500 |
| Reference:
|
[10] Dijkstra J.J., van Mill J., Mogilski J.: The space of infinite-dimensional compact spaces and other topological copies of $(\ell _f^2)^ømega $.Pac. J. Math. 152 (1992), 255-273. MR 1141795 |
| . |
