The Suslinian number and other cardinal invariants of continua

• Autorzy: T. Banakh , V. V. Fedorchuk , J. Nikiel , M. Tuncali
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 54F50: Spaces of dimension ≤ 1 ; curves, dendrites
• 54C05: Continuous maps
• 54F15: Continua and generalizations
• 54F05: Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2010
• Tom: 209
• Numer: 1
• Strony: 43-57

Daty

wydano

2010

Twórcy

autor

T. Banakh

• Uniwersytet Humanistyczno-Przyrodniczy Jana Kochanowskiego, Kielce, Poland
• Department of Mathematics, Ivan Franko Lviv National University, Lviv, Ukraine

autor

V. V. Fedorchuk

• Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Vorob'evy Gory, 1, Moscow, Russia

autor

J. Nikiel

• Instytut Matematyki i Informatyki, Uniwersytet Opolski, Oleska 48, 45-052 Opole, Poland

autor

M. Tuncali

• Nipissing University, North Bay, Ontario, Canada

Identyfikatory

DOI

10.4064/fm209-1-4