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$F_{σ}$ -absorbing sequences in hyperspaces of subcontinua

Helma Gladdines — 1993

Commentationes Mathematicae Universitatis Carolinae

Let $𝒟$ denote a true dimension function, i.e., a dimension function such that $𝒟 (ℝ^{n}) = n$ 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 ${(𝒟_{\geq n} (C (X)))}_{n = 2}^{\infty}$ is $F_{σ}$ -absorbing in $C (X)$ . As a consequence, there is a homeomorphism $h : C (X) \to Q^{\infty}$ such that for all $n$ , $h [{A \in C (X) : 𝒟 (A) \geq n + 1}] = B^{n} \times Q \times Q \times \dots$ , where $B$ denotes the pseudo boundary of the Hilbert cube $Q$ . It follows that if $X$ is a countable infinite product of non-degenerate...

Hyperspaces of infinite-dimensional compacta

Helma Gladdines; Jan Van Mill — 1993

Compositio Mathematica

Hyperspaces of Peano continua of euclidean spaces

Helma Gladdines; Jan van Mill — 1993

Fundamenta Mathematicae

If X is a space then L(X) denotes the subspace of C(X) consisting of all Peano (sub)continua. We prove that for n ≥ 3 the space $L (ℝ^{n})$ is homeomorphic to $B^{\infty}$ , where B denotes the pseudo-boundary of the Hilbert cube Q.