The Suslinian Number and Other Cardinal Invariants of Continua

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 [1]. 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) < 20 , 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 . In this paper we introduce a new cardinal invariant related to the Suslinian property of continua. By a continuum we understand any Hausdorff compact connected space. Following [6], we define a continuuum X to be Suslinian if it contains no uncountable family of pairwise disjoint non-degenerate subcontinua. Suslinian continua were introduced by Lelek [6]. The simplest example of a Suslinian continuum is the usual interval [0, 1]. On the other hand, the existence of non-metrizable Suslinian continua is a subtle problem. The properties of such continua were considered in [1]. It was shown in [1] that each Suslinian continuumX is perfectly normal, rim-metrizable, and 1-dimensional. Moreover, a locally connected Suslinian continuum has weight ≤ ω1. The simplest examples of non-metrizable Suslinian continua are Suslin lines. However this class of examples has a consistency flavour since no Suslin line exists in some models of ZFC (for example, in models satisfying (MA+¬CH) ). It turns out that any example of a non-metrizable locally connected Suslinian continuum necessarily has consistency nature: the existence of such a continuum is equivalent to the existence of a Suslin line, see [1]. This implies that under the Suslin Hypothesis (asserting that no Suslin line exists) each locally connected Suslinian continuum is metrizable. 2000 Mathematics Subject Classification. Primary: 54F15; Secondary: 54C05, 54F05, 54F50.