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Given a metric Peano continuum X we introduce and study the Hölder Dimension Hö-dim(X) = inf{d: there is a 1 d -Hölder onto map f : [0, 1] → X} of X as well as its topological counterpart Hö-dim(X) = inf{Hö-dim(X, d): d is an admissible metric for X}. We show that for each convex metric continuum X the dimension Hö-dim(X) equals the fractal dimension of X. The topological Hölder dimension Hö-dim(M n ) of the n-dimensional universal Menger cube M n equals n. On the other hand, there are 1-dimensional rim-finite Peano continua X with arbitrary prescribed Hö-dim(X) 1.In this paper we are interested in controlled versions of two important topological results: Theorem 0.1 (Alexandroff-Urysohn, 1929). Each metric compact space is a continuous image of the Cantor set.Theorem 0.2 (Hahn-Mazurkiewicz, 1914). Each metric compact connected locally connected space X is a continuous image of the arc [0, 1].In fact, the latter theorem can be derived from the former: given a locally connected metric continuum X first find a continuous surjective map f : C → X from the Cantor set and then use the (local) path-connectedness to extend the map f onto the whole interval [0, 1] ⊃ C.The controlled versions of the Alexandroff-Urysohn and Hahn-Mazurkiewicz Theorems we are interested in treat possible continuity moduli of surjective maps from a Cantor set or [0, 1] onto a given metric space.

Abstract. 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) < 2 ℵ 0 , 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 continuum X is perfectly normal, rim-metrizable, and 1-dimensional. Moreover, a locally connected Suslinian continuum has weight ≤ ω 1 .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.

Let X be a compactum and G an upper semi-continuous decomposition of X such that each element of G is the continuous image of an ordered compactum. If the quotient space X/G is the continuous image of an ordered compactum, under what conditions is X also the continuous image of an ordered compactum? Examples around the (non-metric) HahnMazurkiewicz Theorem show that one must place severe conditions on G if one wishes to obtain positive results. We prove that the compactum X is the image of an ordered compactum when each g ∈ G has 0-dimensional boundary. We also consider the case when G has only countably many non-degenerate elements. These results extend earlier work of the first named author in a number of ways.

We consider those Hausdorff continua S such that each separable subspace of S is hereditarily separable. Due to results of Ostaszewski and Rudin, respectively, all monotonically normal spaces and therefore all continuous Hausdorff images of ordered compacta also have this property. Our study focuses on the structure of such spaces that also possess one of various rim properties, with emphasis given to rim-separability. In so doing we obtain analogues of results of M. Tuncali and I. Lončar, respectively.