E. D. Tymchatyn

On the dimension of certain totally disconnected spaces

It is well known that there exist separable, metrizable, totally disconnected spaces of all dimensions. In this note we introduce the notion of an almost 0-dimensional space and prove that every such space is a totally disconnected subspace of an R-tree and, hence, at most 1-dimensional. As applications we prove that the spaces of homeomorphisms of the universal Menger continua are 1-dimensional and that hereditarily locally connected spaces have dimension at most two.

Continuous images of arcs and inverse limit methods

Introduction Cyclic elements in locally connected continua T-sets in locally connected continua T-maps, T-approximations and continuous images of arcs Inverse sequences of images of arcs

SOME PROPERTIES OF HYPERSPACES WITH APPLICATIONS TO CONTINUA THEORY

1

On minimal maps of 2-manifolds

-dimensional continuous images of arcs Totally regular continua Monotone images

On almost one-to-one maps

\sigma

Weakly confluent mappings and the covering property of hyperspaces

-directed inverse limits References.

The Hahn-Mazurkiewicz theorem for finitely Suslinian continua

1. I n t r o d u c t i o n . In 1972, Lelek introduced the notion of Class (W) in his seminar a t the University of Houston [see below for definitions of concepts mentioned here]. Since then there has been much interest in classifying and characterizing continua in Class (W). F o r example, Cook has a result [5, Theorem 4] which implies t ha t any hereditarily indecomposible cont inuum is in Class (W)} Read [21, Theorem 4] showed tha t all chainable continua are in Class (W), and Feuerbacher proved the following result:

Nöbeling spaces and pseudo-interiors of Menger compacta

We prove that a minimal self-mapping of a compact 2-manifold has tree-like fibers (i.e. all points have preimages which are connected, at most one-dimensional and with trivial shape). We also prove that the only 2-manifolds (compact or not) which admit minimal maps are either finite unions of tori, or finite unions of Klein bottles.

Characterization of Separable Metric R-Trees

A continuous map f: X → Y of topological spaces X, Y is said to be almost 1-to-1 if the set of the points x ∈ X such that f -1 (f(x)) = {x} is dense in X; it is said to be light if pointwise preimages are zero dimensional. We study almost 1-to-1 light maps of some compact and σ-compact spaces (e.g., n-manifolds or dendrites) and prove that in some important cases they must be homeomorphisms or embeddings. In a forthcoming paper we use these results and show that if f is a minimal self-mapping of a 2-manifold M, then point preimages under f are tree-like continua and either M is a union of 2-tori, or M is a union of Klein bottles permuted by f.

A locally connected rim-countable continuum which is the continuous image of no arc

1. Introduction. The notion of Class(H/) (i.e., the class of all continua whichare images of weakly confluent mappings only) was introduced by A. Lelek in1972. Since then several authors have studied these continua and haveattempted to classify them. For example it has been proved that hereditarilyindecomposable continua [1], chainable continua [11] and nonplanar circle-like continua [3] are in Class ( W).In some unpublished work B. Hughes proved that continua with thecovering property of hyperspaces are in Class (W) (see [10, (14.73.21)]) andhe asked whether the converse is true (see [10, (14.73.25)]). Recently, it wasproved that circle-like continua with no local separating subcontinua [4], andtree-like atriodic continua [5] have the covering property, and hence, they arein Class (W). In [5, Theorem 3.1 and Corollary 3.3], a very geometric methodwas introduced in order to check whether certain classes of continua have thecovering property. The following theorem was proved in [4, Theorem 2.2] andit was asked whether its converse is true [4, (5.4)].1.1. Theorem [4, Theorem 2.2]. // A is a continuum which has the covering