Topology of manifolds modeled on countable direct limits of Menger compacta

We construct n-dimensional counterparts of manifolds modeled on the space ?² equipped by the bounded weak topology (-manifolds). For -manifolds we prove the characterization, triangulation and classification theorems. In addition, a universal map of onto Q^∞ (the countable direct limit of Hilbert cubes and Z-embeddings) is constructed and characterized.     

Stratifiable spaces defined by pair collections

We define a pair (F,U) to be a closed set F and an open set U such that F ? U. A sequence of pair collections is used to characterize stratifiable spaces instead of a sequence of neighbornets. We introduce a new class of spaces, called regularly stratifiable spaces, which is defined in terms of pair collections. Every stratifiable μ -space is regularly stratifiable, and every regularly stratifiable space has a σ -almost locally finite base, thus is hereditary M₁. J. Nagata's problem for the dimension of M₁ -spaces is answered positively in the class of regularly stratifiable spaces.     

Preservation of completeness by some continuous maps

We show that a metrizable space Y is completely metrizable if there is a continuous surjection f:X→Y such that the images of open (clopen) subsets of the (0-dimensional paracompact) ?ech-complete space X are resolvable subsets of Y (in particular, e.g., the elements of the smallest algebra generated by open sets in Y).     

Adding machines, kneading maps, and endpoints

Given a unimodal map f, let I=[c₂,c₁] denote the core and set E={(x₀,x₁,…)∈(I,f)|xi∈ω(c,f) for all i∈N}. It is known that there exist strange adding machines embedded in symmetric tent maps f such that the collection of endpoints of (I,f) is a proper subset of E and such that limk→∞Q(k)≠∞, where Q(k) is the kneading map.We use the partition structure of an adding machine to provide a sufficient condition for x to be an endpoint of (I,f) in the case of an embedded adding machine. We then show there exist strange adding machines embedded in symmetric tent maps for which the collection of endpoints of (I,f) is precisely E. Examples of this behavior are provided where limk→∞Q(k) does and does not equal infinity, and in the case where limk→∞Q(k)=∞, the collection of endpoints of (I,f) is always E.     

On universal spaces and absorbing sets related to a transfinite extension of covering dimension

R. Pol has shown that for every countable ordinal number α there exists a universal space for separable metrizable spaces X with trindX?α. W. Olszewski has shown that for every countable limit ordinal number λ there is no universal space for separable metrizable space with trIndX?λ. T. Radul and M. Zarichnyi have proved that for every countable limit ordinal number there is no universal space for separable metrizable spaces with dimWX?α where dimW is a transfinite extension of covering dimension introduced by P. Borst. We prove the same result for another transfinite extension dimC of the covering dimension.As an application, we show that there is no absorbing sets (in the sense of Bestvina and Mogilski) for the classes of spaces X with dimCX?α belonging to some absolute Borel class.     

Transitivities of maps of general topological spaces

In this paper we investigate orbit-transitivity, strong orbit-transitivity, ω-transitivity and open-set-transitivity of maps of general topological spaces. The relation between these transitivities is studied. We discuss various topological spaces, containing pseudo-regular spaces, partially completable spaces, and topological spaces without quasi-isolated points. Several conditions on spaces and on continuity for one transitivity to imply another transitivity are given.     

On metric spaces with the Haver property which are Menger spaces

A metric space (X,d) has the Haver property if for each sequence ?₁,?₂,… of positive numbers there exist disjoint open collections V₁,V₂,… of open subsets of X, with diameters of members of Vi less than ?i and covering X, and the Menger property is a classical covering counterpart to σ-compactness. We show that, under Martin's Axiom MA, the metric square (X,d)×(X,d) of a separable metric space with the Haver property can fail this property, even if X² is a Menger space, and that there is a separable normed linear Menger space M such that (M,d) has the Haver property for every translation invariant metric d generating the topology of M, but not for every metric generating the topology. These results answer some questions by L. Babinkostova [L. Babinkostova, When does the Haver property imply selective screenability? Topology Appl. 154 (2007) 1971-1979; L. Babinkostova, Selective screenability in topological groups, Topology Appl. 156 (1) (2008) 2-9].     

Detecting Hilbert manifolds among isometrically homogeneous metric spaces

We detect Hilbert manifolds among isometrically homogeneous metric spaces and apply the obtained results to recognizing Hilbert manifolds among homogeneous spaces of the form G/H, where G is a metrizable topological group and H is a closed balanced subgroup of G.     

Linearly ordered Eberlein compact spaces

In this note we prove that every Eberlein compact linearly ordered space is metrizable. (By an Eberlein compact space we mean a topological space which can be embedded as a compact subset of a Banach space with the weak topology.)     

Characterizing continuous functions on compact spaces

We consider the following problem: given a set X and a function , does there exist a compact Hausdorff topology on X which makes T continuous? We characterize such functions in terms of their orbit structure. Given the generality of the problem, the characterization turns out to be surprisingly simple and elegant. Amongst other results, we also characterize homeomorphisms on compact metric spaces.     

Inverse limit spaces arising from problems in economics

In this paper we use tools from topology and dynamical systems to analyze the structure of solutions to implicitly defined equations that arise in economic theory, specifically in the study of so-called “backward dynamics”. For this purpose we use inverse limit spaces and shift homeomorphisms to describe solutions which are typical in that they are likely to be observed in future time. These predicted solutions corresponds to attractors in an inverse limit space under the shift homeomorphism(s).     

More stratifiable function spaces

If X is a compact-covering image of a closed subspace of product of a σ-compact Polish space and a compact space, then Ck(X,M), the space of continuous maps of X into M with the compact-open topology, is stratifiable for any metric space M.If X is σ-compact Polish, K is compact and M metric then every point of Ck(X×K,M) has a closure-preserving local base, and hence this function space is M₁.     

Homeomorphisms of unimodal inverse limit spaces with a non-recurrent critical point

Let T be a tent map with the slope strictly between and 2. Suppose that the critical point of T is not recurrent. Let K denote the inverse limit space obtained by using T repeatedly as the bonding map. We prove that every homeomorphism of K to itself is isotopic to some power of the natural shift homeomorphism.     

The hyperspace of hereditarily decomposable subcontinua of a cube is the Hurewicz set

The hyperspaces of hereditarily decomposable continua and of decomposable subcontinua without pseudoarcs in the cube of dimension greater than 2 are homeomorphic to the Hurewicz set of all nonempty countable closed subsets of the unit interval [0,1]. Moreover, in such a cube, all indecomposable subcontinua form a homotopy dense subset of the hyperspace of (nonempty) subcontinua.     

Certain classes of weakly infinite-dimensional spaces and topological games

In [V.V. Fedorchuk, Questions on weakly infinite-dimensional spaces, in: E.M. Pearl (Ed.), Open Problems in Topology II, Elsevier, Amsterdam, 2007, pp. 637-645; V.V. Fedorchuk, Weakly infinite-dimensional spaces, Russian Math. Surveys 42 (2) (2007) 1-52] classes w-m-C of weakly infinite-dimensional spaces, 2?m?∞, were introduced. We prove that all of them coincide with the class wid of all weakly infinite-dimensional spaces in the Alexandroff sense. We show also that transfinite dimensions dimwm, introduced in [V.V. Fedorchuk, Questions on weakly infinite-dimensional spaces, in: E.M. Pearl (Ed.), Open Problems in Topology II, Elsevier, Amsterdam, 2007, pp. 637-645; V.V. Fedorchuk, Weakly infinite-dimensional spaces, Russian Math. Surveys 42 (2) (2007) 1-52], coincide with dimension dimw2=dim, where dim is the transfinite dimension invented by Borst [P. Borst, Classification of weakly infinite-dimensional spaces. I. A transfinite extension of the covering dimension, Fund. Math. 130 (1) (1988) 1-25]. Some topological games which are related to countable-dimensional spaces, to C-spaces, and some other subclasses of weakly infinite-dimensional spaces are discussed.     

The orderability of nonarchimedean spaces

Let X be a nonarchimedean space and C be the union of all compact open subsets of X. The following conditions are listed in increasing order of generality. (Conditions 2 and 3 are equivalent.) 1. X is perfect; 2. C is an F_σ in X; 3. C? is metrizable; 4. X is orderable. It is also shown that X is orderable if $\text{C}\text{?}\text{?C}$ is scattered or X is a GO space with countably many pseudogaps. An example is given of a non-orderable, totally disconnected, GO space with just one pseudogap.     

A splitting property of the upper bounded-Hausdorff convergence

We provide a characterization, based on a splitting property, for the upper bounded-Hausdorff convergence of a net of closed sets. Furthermore, we use this property to describe the local structure of the upper bounded-Hausdorff topology.The author is indebted to S. Levi and R. Lucchetti for continuous suggestions and encouragements.     

Infinite inflations of crumpled cubes

Based upon recent results characterizing Q-manifolds, this paper sets forth an explicit method for retooling certain pathology arising in finite dimensional manifolds as comparable pathology in the Hilbert cube Q. In particular, with reference to an example constructed by the author and J.J. Walsh, it presents an upper semicontinuous decomposition G of Q into points and a null sequence of cellular arcs such that the associated decomposition space is not a Q-manifold, and it also provides a new procedure for embedding finite dimensional compacta as wild subsets of Q.