| Popis: |
Let α be an involution of a Peano continuum X with nowhere dense fixed point set. Let α ⁎ be the induced involution on the hyperspace 2 X of nonempty closed subsets of X topologized by a Hausdorff metric. Let E ⊆ 2 X be a non-degenerate, α ⁎ -invariant hyperspace of X that is an inclusion or growth hyperspace in the sense of Curtis and Schori, and assume that the complement of { X } in E is contractible. Let S ( E ) be its fixed point set. If E is an inclusion hyperspace, then the restriction α ˆ ⁎ of α ⁎ to E is conjugate with the involution i d × τ of the Hilbert cube S ( E ) × Π i ≥ 1 I i , where τ is the involution of Π i ≥ 1 I i that reflects each coordinate across its mid-point. If E is a growth hyperspace of X and X contains no open subset homeomorphic to an arc, then the same result holds. In either case, if the complement of { X } in S ( E ) is contractible, then α ˆ ⁎ is conjugate with the involution σ of Π i ≥ 1 I i that reflects each even coordinate across its mid-point. |
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