The Suslinian number and other cardinal invariants of continua

T. Banakh, et al. "The Suslinian number and other cardinal invariants of continua." Fundamenta Mathematicae 209.1 (2010): 43-57. <http://eudml.org/doc/282760>.

@article{T2010,
abstract = {By the Suslinian number Sln(X) of a continuum X we understand the smallest cardinal number κ such that X contains no disjoint family ℂ of non-degenerate subcontinua of size |ℂ| > κ. For a compact space X, Sln(X) is the smallest Suslinian number of a continuum which contains a homeomorphic copy of X. Our principal result asserts that each compact space X has weight ≤ Sln(X)⁺ and is the limit of an inverse well-ordered spectrum of length ≤ Sln(X)⁺, consisting of compacta with weight ≤ Sln(X) and monotone bonding maps. Moreover, w(X) ≤ Sln(X) if no Sln(X)⁺-Suslin tree exists. This implies that under the Suslin Hypothesis all Suslinian continua are metrizable, which answers a question of Daniel et al. [Canad. Math. Bull. 48 (2005)]. On the other hand, the negation of the Suslin Hypothesis is equivalent to the existence of a hereditarily separable non-metrizable Suslinian continuum. If X is a continuum with $Sln(X) < 2^\{ℵ₀\}$, then X is 1-dimensional, has rim-weight ≤ Sln(X) and weight w(X) ≥ Sln(X). Our main tool is the inequality w(X) ≤ Sln(X)·w(f(X)) holding for any light map f: X → Y.},
author = {T. Banakh, V. V. Fedorchuk, J. Nikiel, M. Tuncali},
journal = {Fundamenta Mathematicae},
keywords = {Suslinian continua; Suslinian number; inverse limits; locally connected continuum; light mappings},
language = {eng},
number = {1},
pages = {43-57},
title = {The Suslinian number and other cardinal invariants of continua},
url = {http://eudml.org/doc/282760},
volume = {209},
year = {2010},
}

TY - JOUR
AU - T. Banakh
AU - V. V. Fedorchuk
AU - J. Nikiel
AU - M. Tuncali
TI - The Suslinian number and other cardinal invariants of continua
JO - Fundamenta Mathematicae
PY - 2010
VL - 209
IS - 1
SP - 43
EP - 57
AB - By the Suslinian number Sln(X) of a continuum X we understand the smallest cardinal number κ such that X contains no disjoint family ℂ of non-degenerate subcontinua of size |ℂ| > κ. For a compact space X, Sln(X) is the smallest Suslinian number of a continuum which contains a homeomorphic copy of X. Our principal result asserts that each compact space X has weight ≤ Sln(X)⁺ and is the limit of an inverse well-ordered spectrum of length ≤ Sln(X)⁺, consisting of compacta with weight ≤ Sln(X) and monotone bonding maps. Moreover, w(X) ≤ Sln(X) if no Sln(X)⁺-Suslin tree exists. This implies that under the Suslin Hypothesis all Suslinian continua are metrizable, which answers a question of Daniel et al. [Canad. Math. Bull. 48 (2005)]. On the other hand, the negation of the Suslin Hypothesis is equivalent to the existence of a hereditarily separable non-metrizable Suslinian continuum. If X is a continuum with $Sln(X) < 2^{ℵ₀}$, then X is 1-dimensional, has rim-weight ≤ Sln(X) and weight w(X) ≥ Sln(X). Our main tool is the inequality w(X) ≤ Sln(X)·w(f(X)) holding for any light map f: X → Y.
LA - eng
KW - Suslinian continua; Suslinian number; inverse limits; locally connected continuum; light mappings
UR - http://eudml.org/doc/282760
ER -