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Making holes in the cone, suspension and hyperspaces of some continua

José G. Anaya, Enrique Castañeda-Alvarado, Alejandro Fuentes-Montes de Oca, Fernando Orozco-Zitli (2018)

Commentationes Mathematicae Universitatis Carolinae

A connected topological space Z is unicoherent provided that if Z = A ∪ B where A and B are closed connected subsets of Z , then A ∩ B is connected. Let Z be a unicoherent space, we say that z ∈ Z makes a hole in Z if Z - { z } is not unicoherent. In this work the elements that make a hole to the cone and the suspension of a metric space are characterized. We apply this to give the classification of the elements of hyperspaces of some continua that make them hole.

Mappings on the dyadic solenoid

Jan M. Aarts, Robbert J. Fokkink (2003)

Commentationes Mathematicae Universitatis Carolinae

Answering an open problem in [3] we show that for an even number k , there exist no k to 1 mappings on the dyadic solenoid.

Maps with dimensionally restricted fibers

Vesko Valov (2011)

Colloquium Mathematicae

We prove that if f: X → Y is a closed surjective map between metric spaces such that every fiber f - 1 ( y ) belongs to a class S of spaces, then there exists an F σ -set A ⊂ X such that A ∈ S and d i m f - 1 ( y ) ∖ A = 0 for all y ∈ Y. Here, S can be one of the following classes: (i) M: e-dim M ≤ K for some CW-complex K; (ii) C-spaces; (iii) weakly infinite-dimensional spaces. We also establish that if S = M: dim M ≤ n, then dim f ∆ g ≤ 0 for almost all g ∈ C ( X , n + 1 ) .

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