Murat Tuncali

Hereditarily indecomposable inverse limits of graphs: shadowing, mixing and exactness

We provide a method of construction of topologically mixing maps f on topological graph G with the shadowing property and such that the inverse limit with f as the single bonding map is a hereditarily indecomposable continuum. Additionally, f can be obtained as an arbitrarily small perturbation of any given topologically exact map on G, and if G is the unit circle, then f is necessarily topologically exact.

Topology in North Bay: Some problems in continuum theory, dimension theory and selections

Publisher Summary This chapter discusses problems in dimension theory, selections and continuum theory. All spaces are assumed to be metrizable and separable, if not stated otherwise. The group G denotes an Abelian group. K ( G , n ) is the Eilenberg–Mac Lane complex, that is, a CW complex such that π n ( K ( G , n )) ≈ G and π i ( K ( G , n )) ≈ 0 for all i ≠ n . The cohomological dimension of a space X with respect to the coefficient group G is denoted by dim G X . A space Y is an absolute (neighborhood) extensor for X [notation: Y ∈ A ( N ) E ( X )] if any map to Y , defined on an arbitrary closed subspace A of X , can be extended to a map of the whole X to Y (resp., to a map of some open neighborhood of A to Y ). Section 2 of this chapter discusses the problems on extension dimension. In the third section, problems concerning selections and C -spaces are discussed. Section 4 discusses questions concerning the parametric version of disjoint disks property. The last section is devoted to locally connected Hausdorff continua and rim-metrizablity.

Homogeneous Suslinian Continua

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.

Plane continua and totally disconnected sets of buried points

It is shown that there is a continuum in the plane whose set of buried points is totally disconnected and weakly 1-dimensional, but not zerodimensional. This answers a problem of Curry, Mayer and Tymchatyn.