On uncountable collections of continua and their span

Dušan Repovš; Arkadij Skopenkov; Evgenij Ščepin

Colloquium Mathematicae (1996)

  • Volume: 69, Issue: 2, page 289-296
  • ISSN: 0010-1354

Abstract

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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.

How to cite

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Repovš, Dušan, Skopenkov, Arkadij, and Ščepin, Evgenij. "On uncountable collections of continua and their span." Colloquium Mathematicae 69.2 (1996): 289-296. <http://eudml.org/doc/210342>.

@article{Repovš1996,
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.},
author = {Repovš, Dušan, Skopenkov, Arkadij, Ščepin, Evgenij},
journal = {Colloquium Mathematicae},
keywords = {uncountable collection of compacta; deleted product; chainable continua; span; equivariant maps; symmetric span; tree-like continuum},
language = {eng},
number = {2},
pages = {289-296},
title = {On uncountable collections of continua and their span},
url = {http://eudml.org/doc/210342},
volume = {69},
year = {1996},
}

TY - JOUR
AU - Repovš, Dušan
AU - Skopenkov, Arkadij
AU - Ščepin, Evgenij
TI - On uncountable collections of continua and their span
JO - Colloquium Mathematicae
PY - 1996
VL - 69
IS - 2
SP - 289
EP - 296
AB - 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.
LA - eng
KW - uncountable collection of compacta; deleted product; chainable continua; span; equivariant maps; symmetric span; tree-like continuum
UR - http://eudml.org/doc/210342
ER -

References

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