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1 Fundamenta Mathematicae

1 2010

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The Suslinian number and other cardinal invariants of continua

100%

Banakh T. , Fedorchuk V. , Nikiel J. , Tuncali M.

nr 1

43-57

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.