Elusive Examples of Non-Metrizable Continua which Admit a Whitney Map

The main purpose of this paper is to prove that the class of near locally connected continua contains no non-metrizable continuum 𝑋 which admits a Whitney map for 𝐶(𝑋). In particular, each near locally connected continuum 𝑋 which admits a Whitney map for 𝐶(𝑋) is metrizable.

ENDS OF CUSP-UNIFORM GROUPS OF LOCALLY CONNECTED CONTINUA — I

Due to works by Bestvina–Mess, Swarup and Bowditch, we now have complete knowledge of how splittings of a word-hyperbolic group G as a graph of groups with finite or two-ended edge groups relate to the cut point structure of its boundary. It is central in the theory that ∂G is a locally connected continuum (a Peano space). Motivated by the structure of tight circle packings, we propose to generalize this theory to cusp-uniform groups in the sense of Tukia. A Peano space X is cut-rigid, if X has no cut point, no points of infinite valence and no cut pairs consisting of bivalent points. We prove: Theorem. Suppose X is a cut-rigid space admitting a cusp-uniform action by an infinite group. If X contains a minimal cut triple of bivalent points, then there exists a simplicial tree T, canonically associated with X, and a canonical simplicial action of Homeo(X) on T such that any infinite cusp-uniform group G of X acts cofinitely on T, with finite edge stabilizers. In particular, if X is such that T is locally finite, then any cusp-uniform group G of X is virtually free.

A note on periodic points and recurrent points of maps of dendrites

Let f: X → X be a map of a continuum X. Let P(f) denote the set of all periodic points of f and R(f) denote the set of all recurrent points of f. In [2], Coven and Hedlund proved that if f: I → I is a map of the unit interval I = [0, 1], then CI(P(f)) = CI(R(f)). In [7], Ye generalised this result to maps of a tree. It is natural to ask whether the result generalises to maps of a dendrite. (A dendrite is a locally connected continuum which contains no simple closed curve.) The aim of this paper is to show that the answer is negative.

Separators in Continuous Images of Ordered Continua and Hereditarily Locally Connected Continua

Let X be a Hausdorff space which is the continuous image of an ordered continuum. We prove that every irreducible separator of X is metrizable. This is a far reaching extension of the 1967 theorem of S. Mardešić which asserts that X has a basis of open sets with metrizable boundaries. Our first result is then used to show that, in particular, if Y is an hereditarily locally connected continuum, then for subsets of Y quasi-components coincide with components, and that the boundary of each connected open subset of Y is accessible by ordered continua. These results answer open problems in the literature due to the fourth and third authors, respectively.

A property of hereditarily locally connected continua related to arcwise accessibility

A continuum (that is, a compact connected Hausdorff space) is hereditarily locally connected if each of its subcontinua is locally connected. It is shown that a continuum X is hereditarily locally connected if and only if for each connected open set U in X and each point p in the boundary of U, U ∪ {p} is locally connected. This result is used to prove that if X is an hereditarily locally connected continuum, U is a connected open subset of X, p is an element of the boundary of U and X is first countable at p, then p is arcwise accessible from U.

A Note On Arc-Preserving Functions For Manifolds1

Hall and Puckett [2] have shown that an arc - preserving function defined on a locally connected continuum having no local separating points is a homeomorphism if its total image is not an arc or point. This note shows that their results can be extended to non-compact manifolds.