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 , 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm209-1-4,
author = {T. Banakh and V. V. Fedorchuk and J. Nikiel and M. Tuncali},
title = {The Suslinian number and other cardinal invariants of continua},
journal = {Fundamenta Mathematicae},
volume = {209},
year = {2010},
pages = {43-57},
zbl = {1260.54050},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm209-1-4}
}
T. Banakh; V. V. Fedorchuk; J. Nikiel; M. Tuncali. The Suslinian number and other cardinal invariants of continua. Fundamenta Mathematicae, Tome 209 (2010) pp. 43-57. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm209-1-4/