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Hyperspaces of generalized continua which are infinite cylinders. (English) Zbl 1429.54040

A continuum is a compact connected metric space. A generalized continuum is a locally compact, non-compact, separable, connected and metrizable space. If \(X\) is either a continuum or a generalized continuum, the hyperspace of subcontinua of \(X\) is denoted by \(C(X)\) and is endowed with the Hausdorff metric.
If \(X\) is a continuum, \(C(X)\) has the cone=hyperspace property if there exists a homeomorphism \(h:C(X)\rightarrow \text{cone}(X)\) such that \(h(\{x\})=(x,0)\) for each \(x\in X\) and \(h(X)=\) vertex of cone\((X)\).
Continua \(X\) for which \(C(X)\) has the cone=hyperspace property have been largely studied since the 1970’s. A very complete account of what is known on this topic can be found in [A. Illanes and S. B. Nadler Jr., Hyperspaces: fundamentals and recent advances. New York, NY: Marcel Dekker (1999; Zbl 0933.54009)].
In the paper under review, the authors define the natural extension of this concept for the case of generalized continua as follows: a generalized continuum is in class Cyl if there exists a homeomorphism \(h:C(X)\rightarrow X\times \mathbb{R}_{\geq 0}\) such that \(h({x})=(x,0)\) for each \(x\in X\).
The authors show that many of the results for the compact case can be naturally extended to generalized continua. So, they work on three lines.
– Equivalences. They prove a characterization for a generalized continuum to be in Cyl in terms of selections and compactwise Whitney maps, similar to the one given by the reviewer for the cone=hyperspace property.
– Sufficient conditions. Among other results they prove that if \(X\) is a \(1\)-dimensional generalized continuum with the Kelley property and such that all of its constituants are homeomorphic to the real line, then \(X\) is in Cyl. As a corollary of this result, they obtain that there exist uncountably many hereditarily decomposable continua in Cyl. This result is in contrast with Nadler’s result for the compact case that says that there are only eight hereditarily decomposable continua \(X\) for which \(C(X)\) is homeomorphic to cone\((X)\).
– Obstructions to being in the class Cyl. They show that, as in the metric case, if a generalized continuum \(X\) is of type \(N\) at some of its subcontinua or if \(X\) contains an \(R^{3}\)-point, then \(X\) does not belong to the class Cyl.

MSC:

54F16 Hyperspaces of continua
54C10 Special maps on topological spaces (open, closed, perfect, etc.)

Citations:

Zbl 0933.54009
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References:

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