Fans having unique cones
A continuum is a compact connected nondegenerate metric space. We say that a continuum \(X\) has unique cone provided that the following property holds: if \(Y\) is a continuum and \(cone(X)\) is homeomorphic to \(cone(Y)\), then \(X\) is homeomorphic to \(Y\). Now, a fan is an arcwise continuum with exactly one ramification point having the property that the intersection of any two subcontinua is connected. In [Topol. Proc. 52, 35--43 (2018; Zbl 1409.54011)], \textit{D. Michalik} proved the following result: Theorem. Let us assume that \(X\) and \(Y\) are locally connected curves not being ANRs. Then \(cone(X)\) and \(cone(Y)\) are homeomorphic if and only if \(X\) is homeomorphic to \(Y\). The objective of this paper under review is to prove that a fan has unique cone if and only if it is not the cone over a zero-dimensional non-degenerate compact metric space.
scientific article; zbMATH DE number 7401497
scientific article