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

Gladdines, Helma

Commentationes Mathematicae Universitatis Carolinae, Tome 34 (1993) no. 4, pp. 729-745.

Résumé

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.

Détail

Classification : 54B20, 54F15, 54F45, 55M10, 57N20
Keywords: Hilbert cube; absorbing system; $F_\sigma$; $F_{\sigma \delta}$; capset; Peano continuum; hyperspace; hyperspace of subcontinua; covering dimension; cohomological dimension