On uncountable collections of continua and their span
Volume 69 / 1996
Colloquium Mathematicum 69 (1996), 289-296
DOI: 10.4064/cm-69-2-289-296
Abstract
We prove that if the Euclidean plane $ℝ^2$ contains an uncountable collection of pairwise disjoint copies of a tree-like continuum X, then the symmetric span of X is zero, sX = 0. We also construct a modification of the Oversteegen-Tymchatyn example: for each ε > 0 there exists a tree $X ⊂ ℝ^2$ such that σX < ε but X cannot be covered by any 1-chain. These are partial solutions of some well-known problems in continua theory.
