Jacek Nikiel

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

The Hahn-Mazurkiewicz theorem for hereditarily locally connected continua

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Orderability properties of a zero-dimensional space which is a continuous image of an ordered compactum

-dimensional continuous images of arcs Totally regular continua Monotone images

On Suslinian Continua

\sigma

Homogeneous Suslinian Continua

-directed inverse limits References.

Images of arcs - a nonseparable version of the Hahn-Mazurkiewicz theorem

Abstract It is proved that each hereditarily locally connected continuum is a continuous image of an arc and is a rational space.

Topologies on pseudo-trees and applications

Abstract If X is a zero-dimensional Hausdorff space which is a continuous image of a compact linearly ordered topological space, then X can be embedded into some dendron. If, moreover, X is separable then X can be embedded into some arc. Both results can be treated as orderability theorems.

On the rim-structure of continuous images of ordered compacta.

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 omega1 and under appropriate set-theoretic conditions are metrizable. Non-separable locally connected Suslinian continua are rim-finite on some open set

SEPARABLE ZERO-DIMENSIONAL SPACES WHICH ARE CONTINUOUS IMAGES OF ORDERED COMPACTA

A continuum is said to be Suslinian if it does not contain uncountably many mutually exclusive non-degenerate subcontinua. Fitzpatrick and Lelek have shown that a metric Suslinian continuum X has the property that the set of points at which X is connected im kleinen is dense in X. We extend their result to Hausdorff Suslinian continua and obtain a number of corollaries. In particular, we prove that a homogeneous, non-degenerate, Suslinian continuum is a simple closed curve and that each separable, non-degenerate, homogenous, Suslinian continuum is metrizable. Lamar University, Department of Mathematics, Beaumont, TX, U.S.A. e-mail: dale.daniel@lamar.edu Opole University, Institute of Mathematics and Informatics, Opole, Poland e-mail: nikiel@math.uni.opole.pl Texas AM revised September 11, 2008. Published electronically January 26, 2011. The fourth and the fifth authors are partially supported by National Science and Engineering Research Council of Canada grants No:141066-2000 and No:OGP0005616, respectively. AMS subject classification: 54F15, 54C05, 54F05, 54F50.