Homeomorphisms of inverse limits of metric spaces
Homogeneity and rigidity in Erdös spaces
The classical Erdös spaces are obtained as the subspaces of real separable Hilbert space consisting of the points with all coordinates rational or all coordinates irrational, respectively. One can create variations by specifying in which set each coordinate is allowed to vary. We investigate the homogeneity of the resulting subspaces. Our two main results are: in case all coordinates are allowed to vary in the same set the subspace need not be homogeneous, and by specifying different sets for different...
Homogeneous cohomology manifolds which are inverse limits
Homotopically fixed arcs and the contractibility of dendroids
Homotopy properties of curves
Conditions are investigated that imply noncontractibility of curves. In particular, a plane noncontractible dendroid is constructed which contains no homotopically fixed subset. A new concept of a homotopically steady subset of a space is introduced and its connections with other related concepts are studied.
Homotopy properties of pathwise connected continua
Homotopy separators and mappings into cubes
Homotopy types of one-dimensional Peano continua
Let X and Y be one-dimensional Peano continua. If the fundamental groups of X and Y are isomorphic, then X and Y are homotopy equivalent. Every homomorphism from the fundamental group of X to that of Y is a composition of a homomorphism induced from a continuous map and a base point change isomorphism.
Hurewicz-Serre theorem in extension theory
The paper is devoted to generalizations of the Cencelj-Dranishnikov theorems relating extension properties of nilpotent CW complexes to their homology groups. Here are the main results of the paper: Theorem 0.1. Let L be a nilpotent CW complex and F the homotopy fiber of the inclusion i of L into its infinite symmetric product SP(L). If X is a metrizable space such that for all k ≥ 1, then and for all k ≥ Theorem 0.2. Let X be a metrizable space such that dim(X) < ∞ or X ∈ ANR. Suppose...
Hyperspace retractions for curves [Book]
Hyperspace selections avoiding points
We deal with a hyperspace selection problem in the setting of connected spaces. We present two solutions of this problem illustrating the difference between selections for the nonempty closed sets, and those for the at most two-point sets. In the first case, we obtain a characterisation of compact orderable spaces. In the latter case --- that of selections for at most two-point sets, the same selection property is equivalent to the existence of a ternary relation on the space, known as a cyclic...
Hyperspaces of Peano continua are Hubert cubes
Hyperspaces of Peano continua of euclidean spaces
If X is a space then L(X) denotes the subspace of C(X) consisting of all Peano (sub)continua. We prove that for n ≥ 3 the space is homeomorphic to , where B denotes the pseudo-boundary of the Hilbert cube Q.
Hyperspaces of polyhedra are Hilbert cubes
Hyperspaces of two-dimensional continua
Let X be a compact metric space and let C(X) denote the space of subcontinua of X with the Hausdorff metric. It is proved that every two-dimensional continuum X contains, for every n ≥ 1, a one-dimensional subcontinuum with . This implies that X contains a compact one-dimensional subset T with dim C (T) = ∞.
Hyperspaces of universal curves and 2-cells are true -sets
It is shown that the following hyperspaces, endowed with the Hausdorff metric, are true absolute -sets: (1) ℳ ²₁(X) of Sierpiński universal curves in a locally compact metric space X, provided ℳ ²₁(X) ≠ ∅ ; (2) ℳ ³₁(X) of Menger universal curves in a locally compact metric space X, provided ℳ ³₁(X) ≠ ∅ ; (3) 2-cells in the plane.
Hyperspaces of various locally connected ѕubcontinua
Ideals of uniformly continuous mappings on pseudometric spaces
If it looks and smells like the reals...
Given a topological space ⟨X,T⟩ ∈ M, an elementary submodel of set theory, we define to be X ∩ M with topology generated by U ∩ M:U ∈ T ∩ M. We prove that if is homeomorphic to ℝ, then . The same holds for arbitrary locally compact uncountable separable metric spaces, but is independent of ZFC if “local compactness” is omitted.
Images of arcs - a nonseparable version of the Hahn-Mazurkiewicz theorem