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On uncountable collections of continua and their span

Dušan RepovšArkadij SkopenkovEvgenij Ščepin — 1996

Colloquium Mathematicae

We prove that if the Euclidean plane 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 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.

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