$F_{\sigma }$-absorbing sequences in hyperspaces of subcontinua

Gladdines, Helma. "$F_\sigma $-absorbing sequences in hyperspaces of subcontinua." Commentationes Mathematicae Universitatis Carolinae 34.4 (1993): 729-745. <http://eudml.org/doc/247511>.

@article{Gladdines1993,
abstract = {Let $\mathcal \{D\}$ denote a true dimension function, i.e., a dimension function such that $\mathcal \{D\}(\mathbb \{R\}^n) = n$ 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 $(\mathcal \{D\}_\{\ge 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[\lbrace A \in C(X) : \mathcal \{D\}(A) \ge n+1\rbrace ] = 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 $\mathcal \{D\}_\{\ge n\}(C(X))$ is an $F_\sigma $-absorber (capset) for $C(X)$, for every $n \ge 2$. 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 $k \in \mathbb \{N\}$, the sequence $(\operatorname\{dim\}_\{\ge 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.},
author = {Gladdines, Helma},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Hilbert cube; absorbing system; $F_\sigma $; $F_\{\sigma \delta \}$; capset; Peano continuum; hyperspace; hyperspace of subcontinua; covering dimension; cohomological dimension; countable infinite product of non-degenerate Peano continua; hyperspace of all subcontinua; pseudo-boundary of the Hilbert cube; covering dimension; cohomological dimension},
language = {eng},
number = {4},
pages = {729-745},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {$F_\sigma $-absorbing sequences in hyperspaces of subcontinua},
url = {http://eudml.org/doc/247511},
volume = {34},
year = {1993},
}

TY - JOUR
AU - Gladdines, Helma
TI - $F_\sigma $-absorbing sequences in hyperspaces of subcontinua
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1993
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 34
IS - 4
SP - 729
EP - 745
AB - Let $\mathcal {D}$ denote a true dimension function, i.e., a dimension function such that $\mathcal {D}(\mathbb {R}^n) = n$ 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 $(\mathcal {D}_{\ge 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[\lbrace A \in C(X) : \mathcal {D}(A) \ge n+1\rbrace ] = 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 $\mathcal {D}_{\ge n}(C(X))$ is an $F_\sigma $-absorber (capset) for $C(X)$, for every $n \ge 2$. 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 $k \in \mathbb {N}$, the sequence $(\operatorname{dim}_{\ge 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.
LA - eng
KW - Hilbert cube; absorbing system; $F_\sigma $; $F_{\sigma \delta }$; capset; Peano continuum; hyperspace; hyperspace of subcontinua; covering dimension; cohomological dimension; countable infinite product of non-degenerate Peano continua; hyperspace of all subcontinua; pseudo-boundary of the Hilbert cube; covering dimension; cohomological dimension
UR - http://eudml.org/doc/247511
ER -