RETRACTED Another type of quotients of hyperspaces
Keywords:
Absolute retract, continuum, decomposable continuum, dendroid, fan, hereditarily decomposable continuum, hereditarily indecomposable continuum, hereditarily unicoherent continuum, hyperspaces, indecomposable continuum, induced map, n-fold suspension hyperspace, property of Kelley, smooth fan, trivial shape, unicoherent continuumAbstract
Let X be a continuum, and let n be a positive integer. We introduce a new type of quotient of hyperspaces. For this, consider the hyperspace 2X of X, consisting of all nonempty closed subsets of X, and the n-fold hyperspace of X, Cn(X), whose elements are all nonempty closed subsets of X with at most n components. These hyperspaces are topologized with the Hausdorff metric. We define the quotient space 2Xn = 2X/Cn(X), with the quotient topology. We call 2Xn the n-fold suspension hyperspace of X. Note that this is the first time that a quotient of the hyperspace 2X has been taken. We prove several properties of n-fold suspension hyperspaces. For example: we show that 2Xn is a unicoherent continuum. We give sufficient conditions to have that 2Xn is contractible. We prove that the continuum X is locally connected if and only if 2X n is the Hilbert cube. Given a map f : X ! Y between continua, we define and study the corresponding induced map between the n-fold suspension hyperspaces of X and Y .
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