On Suslinian Continua

Abstract

Abstract. A continuum is said to be Suslinian if it does not contain uncountably many mutually exclusive nondegenerate subcontinua. We prove that Suslinian continua are perfectly normal and rim-metrizable. Locally connected Suslinian continua have weight at most ω 1 and under appropriate set-theoretic conditions are metrizable. Non-separable locally connected Suslinian continua are rim-finite on some open set.

The Suslinian number and other cardinal invariants of continua

“…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.…”

“…Embed Y into the Tychonov cube [0, 1] κ where κ = w(Y ). It follows from the Tietze-Urysohn Theorem that the map f can be extended to a mapf : Z → [0, 1] κ . Observe that each non-empty open set U ⊂ Z has no one-point component (otherwise this one-point component would be a quasi-component and consequently Z would contain a non-trivial clopen subset which contradicts the connectedness of Z).…”

“…Assuming that the family C fails to separate the points x, y, we will get that for each neigborhood W ⊂ [0, 1] κ of z there is a subcontinuum K W ⊂ f −1 (W ) containing both x and y. The net (K W ) W has a limit point K in the hyperspace exp(Z) of Z (the latter means that for any neighborhood O(K) of K in exp(Z) and any neighborhood W of z in [0,1] κ there is a neighborhood V ⊂ W of z such that K V ∈ O(K)). This limit point K is a subcontinuum of Z containing the points x and y and lying in f −1 (z) which is impossible as the latter set is zero-dimensional.…”

“…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.…”

“…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.…”