On the (n,m)-fold hyperspace suspension of a continuum
Date
2023-07-03
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Benemérita Universidad Autónoma de Puebla
Abstract
"Throughout the years, the study of hyperspaces has acquired a notorious importance within the theory of continua. Recall that a continuum X is a nonempty connected, compact and metric space, and a hyperspace of a continuum is family of closed subsets of X sharing certain properties. Introduced the hyperspace suspension of a continuum X as the quotient space C1(X)/F1(X), denoted by HS(X). Later, introduced the n-fold hyperspace suspension of a continuum X, for each n 2 N and n2 as the quotient space Cn(X)/Fn(X), denoted by HSn(X), where there are described some general properties about this hyperspace. As usual, once there is a new hyperspace around, the natural question about the uniqueness arises. We say that a continuum X has unique hyperspace K(X) if for any continuum Y such that K(X) is homeomorphic to K(Y), then X is homeomorphic to Y. More information about the uniqueness of hyperspaces HSn(X) for some families of continua can. Afterwards, introduced, for n, m 2 N with m=n, the (n,m)-fold hyperspace suspension of a continuum X, denoted by HSn m(X), and defined as the quotient space Cn(X)/Fm(X) obtained from Cn(X) by shrinking Fm(X) to a one-point set with the quotient topology".
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