$F_\sigma $-absorbing sequences in hyperspaces of subcontinua
Gladdines, Helma
Commentationes Mathematicae Universitatis Carolinae, Tome 34 (1993), p. 729-745 / Harvested from Czech Digital Mathematics Library
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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.

Publié le : 1993-01-01
Classification:  54B20,  54F15,  54F45,  55M10,  57N20 
@article{118630,
     author = {Helma Gladdines},
     title = {$F\_\sigma $-absorbing sequences in hyperspaces of subcontinua},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {34},
     year = {1993},
     pages = {729-745},
     zbl = {0813.57020},
     mrnumber = {1263802},
     language = {en},
     url = {http://dml.mathdoc.fr/item/118630}
}

Gladdines, Helma. $F_\sigma $-absorbing sequences in hyperspaces of subcontinua. Commentationes Mathematicae Universitatis Carolinae, Tome 34 (1993) pp. 729-745. http://gdmltest.u-ga.fr/item/118630/

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