Helma Gladdines
$F_\sigma $-absorbing sequences in hyperspaces of subcontinua

Comment.Math.Univ.Carolinae 34,4 (1993) 729-745.

Abstract: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 $n$, $h[\{A \in C(X) : \Cal D(A) \geq n+1\}] = B^n \times Q \times Q \times...$, 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 $n \geq 2$. \par Let $dim$ 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 $(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.

Keywords: Hilbert cube, absorbing system, $F_\sigma $, $F_{\sigma \delta }$, capset, Peano continuum, hyperspace, hyperspace of subcontinua, covering dimension, cohomological dimension
AMS Subject Classification: 57N20