On hyperspaces of polyhedra
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Abstract:
Let $Q = {[ - 1,1]^\omega }$ be the Hilbert cube and \[ {Q_f} = \left \{ {({x_i}) \in Q|{x_i} = 0{\text {except for finitely many }}i} \right \}.\] For a compact connected polyhedron $X$ with $\dim X > 0$, the hyperspaces of (nonempty) subcompacta, subcontinua, and compact subpolyhedra of $X$ are denoted by ${2^X},C(X)$, and ${\text {Pol(}}X{\text {)}}$, respectively. And let ${C^{{\text {Pol}}}}(X) = C(X) \cap {\text {Pol(}}X{\text {)}}$. It is shown that the pair $({2^X},{\text {Pol(}}X{\text {)}})$ is homeomorphic to $(Q,{Q_f})$. In case $X$ has no free arc, it is also proved that the pair $(C(X),{C^{{\text {Pol}}}}(X))$ is homeomorphic to $(Q,{Q_f})$.References
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Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 110 (1990), 1089-1097
- MSC: Primary 57N20; Secondary 54B20
- DOI: https://doi.org/10.1090/S0002-9939-1990-1037223-0
- MathSciNet review: 1037223