Embedding cones in hyperspaces (Q1935838)

The author presents results about continua \(X\) that are cone-embeddable in \(C(X)\). For a metric continuum \(X\), \(C(X)\) denotes the hyperspace of subcontinua of \(X\) and \(Cone(X)\) denotes the topological cone of \(X\). A continuum \(X\) is said to be cone-embeddable in \(C(X)\) provided that there exists an embedding \(h\) from \(Cone(X)\) to \(C(X)\) such that \(h(x,0) = \{x\}\) for each \(x \in X\). The main results of the paper are: - A cone-embeddable continuum contains at most one terminal subcontinuum, in that case the terminal subcontinuum is also a cone-embeddable continuum. - If \(X\) is a continuum cone-embeddable in \(C(X)\), then \(C(X)\) is contractible. - The product of finitely many cone-embeddable continua is a cone-embeddable continuum. - If \(X\) is a dendrite then \(X\) is an (ordered) cone-embeddable continuum.