On resolvable primal spaces

A topological space is called resolvable if it is a union of two disjoint dense subsets, and is n-resolvable if it is a union of n mutually disjoint dense subsets. Clearly a resolvable space has no isolated points. If f is a selfmap on X, the sets A?X with f (A)?A are the closed sets of an Alexandroff topology called the primal topology 𝒫(f ) associated with f. We investigate resolvability for primal spaces (X, 𝒫(f)). Our main result is that an Alexandroff space is resolvable if and only if it has no isolated points. Moreover, n-resolvability and other related concepts are investigated for primal spaces.     

Resolution property in generalized k-spaces

We show that a T₁ space X is resolvable if the set of limit points λ (X) of various simultaneously separated subsets of X is dense in X. Moreover, if λ (X) is open also, then X is ω-resolvable. It follows that a self-dense, Hausdroff space satisfying a generalized k-space (sequential space) condition is resolvable (respectively, ω-resolvable).     

SMOOTHNESS PROPERTIES OF REAL-VALUED FUNCTIONS BASED ON THEIR OSCILLATION

Abstract

We consider the following two selection principles for topological spaces:

Principle 1: For each sequence of dense subsets, there is a sequence of points from the space, the n-th point coming from the n-th dense set, such that this set of points is dense in the space;

Principle 2: For each sequence of dense subsets, there is a sequence of finite sets, the n-th a subset of the n-th dense set, such that the union of these finite sets is dense in the space.

We show that for separable metric space X one of these principles holds for the space C_p (X) of realvalued continuous functions equipped with the pointwise convergence topology if, and only if, a corresponding principle holds for a special family of open covers of X. An example is given to show that these equivalences do not hold in general for Tychonoff spaces. It is further shown that these two principles give characterizations for two popular cardinal numbers, and that these two principles are intimately related to an infinite game that was studied by Berner and Juhász.     

Resolvability and monotone normality

A space X is said to be κ-resolvable (resp., almost κ-resolvable) if it contains κ dense sets that are pairwise disjoint (resp., almost disjoint over the ideal of nowhere dense subsets). X is maximally resolvable if and only if it is Δ(X)-resolvable, where Δ(X) = min{|G| : G ≠ open}. We show that every crowded monotonically normal (in short: MN) space is ω-resolvable and almost μ-resolvable, where μ = min{2 ^ω , ω ₂}. On the other hand, if κ is a measurable cardinal then there is a MN space X with Δ(X) = κ such that no subspace of X is ω ₁-resolvable. Any MN space of cardinality < ℵ _ω is maximally resolvable. But from a supercompact cardinal we obtain the consistency of the existence of a MN space X with |X| = Δ(X) = ℵ _ω such that no subspace of X is ω ₂-resolvable. The preparation of this paper was supported by OTKA grant no. 61600     

Fuzzy dynamical systems

A fuzzy dynamical system on an underlying complete, locally compact metric state space X is defined axiomatically in terms of a fuzzy attainability set mapping on X. This definition includes as special cases crisp single and multivalued dynamical systems on X. It is shown that the support of such a fuzzy dynamical system on X is a crisp multivalued dynamical system on X, and that such a fuzzy dynamical system can be considered as a crisp dynamical system on a state space of nonempty compact fuzzy subsets of X. In addition fuzzy trajectories are defined, their existence established and various properties investigated.     

Semi-Lipschitz Functions and Best Approximation in Quasi-Metric Spaces

We show that the set of semi-Lipschitz functions, defined on a quasi-metric space (X, d), that vanish at a fixed point x₀X can be endowed with the structure of a quasi-normed semilinear space. This provides an appropriate setting in which to characterize both the points of best approximation and the semi-Chebyshev subsets of quasi-metric spaces. We also show that this space is bicomplete.     

On von neumann's variation of the weierstrass-stone theorem

Let X be a compact HausdorfF space and let D(X) be the set of all continuous real-valued functions f defined on X and such that 0 ≤ f(x) ≤ 1, for all x ? X. The set D(X) is equipped with the uniform topology. We characterize the uniform closure of subsets A ? D(X) containing 0 and 1 and ?ψ + (1 ? ?)η, whenever they contain ?, ψ and η     

On the Weight of Nowhere Dense Subsets in Compact Spaces

We study a new cardinal-valued invariant ndw(X) (calling it the nd-weight of X) of a topological space which is defined as the least upper bound of the weights of nowhere dense subsets of X. The main result is the proof of the inequality hl(X)ndw(X) for compact sets without isolated points ((hl is the hereditary Lindelof number). This inequality implies that a compact space without isolated points of countable nd-weight is completely normal. Assuming the continuum hypothesis, we construct an example of a nonmetrizable compact space of countable nd-weight without isolated points.     

Compact supported endographs and fuzzy sets

A metric is defined on a space of functions from a locally compact metric space X into the unit interval I in terms of the Hausdorff metric distance between their compact supported endographs in X × I. Convergence in this metric is shown to be equivalent to the conjunction of the Hausdorff metric convergence of supports in X and two conditions involving numerical values of the functions. The space of nonempty compact subsets of X with the Hausdorff metric is imbedded in the above function space by the characteristic function on subsets of X. Applications of these results to fuzzy subsets of X and fuzzy dynamical systems on X are indicated.     

The Finite Dimensional Approximation Property and the AR-Property in Needle Point Spaces

We introduce the notion of the finite dimensional approximationproperty (the FDAP) and prove that if a subset X of a linearmetric space has the FDAP, then every non-empty convex subsetof X is an AR. As an application we show that every needle point space X containsa dense linear subspace E with the following properties: (i) E contains a non-empty compact convex set with no extremepoints; (ii) all non-empty convex subsets of E are AR.     

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).     

ech-complete spaces and the upper topology

Let X be a topological space and let be the set of all compact subsets of X. The purpose of this note is to prove the following: if X is regular and q-space, then X is Lindelöf and ech-complete if and only if there exists a continuous map f from a Lindelöf and ech-complete space Y to the space endowed with the upper topology, such that f(Y) is cofinal in . This result extends the following result of Saint Raymond and Christensen: if X is separable metrizable, then X is a Polish space if and only if the space endowed with the Vietoris topology is the continuous image of a Polish space.     

On the Distribution Behaviour of Sequences

This paper presents results concerning those sets of finite Borel measures μ on a locally compact Hausdorff space X with countable topological base which can be represented as the set of limit distributions of some sequence. They arc characterized by being nonanpty, closed, connected and containing only measures μ with μ(X) = 1 (if X is compact) or 0 ≤ μ(X) ≤ 1 (if X is not compact). Any set with this properties can be obtained as the set of limit distributions of a sequence even by rearranging an arbitrarily given sequence which is dense in the sense that the set of accumulation points is the whole space X. The typical case (in the sense of Baire categories) is that a sequence takes as limit distributions all possible measures of this kind. This gives new aspects for the recent theory of maldistribukd sequences.     

Detecting codimension one manifold factors with 0-stitched disks

The 0-stitched disks property is introduced and shown to detect codimension one manifold factors of dimension n?4. It is shown that if a space X is an ANR and has the 0-stitched disks property, then X has the disjoint homotopies property. It follows that if a space X is a resolvable generalized manifold of dimension n?4 with the 0-stitched disks property, then X is a codimension one manifold factor. Whether or not the 0-stitched disks property is equivalent to the disjoint homotopies property remains an open question.     

Metric,Fractal Dimensional and Baire Results on the Distribution of Subsequences

Let X be a locally compact metric space. One important object connected with the distribution behavior of an arbitrary sequence x on X is the set M( x ) of limit measures of x . It is defined as the set of accumulation points of the sequence of the discrete measures induced by x . Using binary representation of reals one gets a natural bijective correspondence between infinite subsets of the set ℕ of positive integers and numbers in the unit interval I = 〈0, 1]. Hence to each sequence x = (x_n)_n∈ℕ ∈ X^ℕ and every a I there corresponds a subsequence denoted by a x . We investigate the set M(a x ) for given x with emphasis on the behavior for “typical” a in the sense of Baire category, Lebesgue measure and Hausdorff dimension.     

Hyperspaces of Normed Linear Spaces with the Attouch–Wets Topology

Let Cld _AW (X) be the hyperspace of nonempty closed subsets of a normed linear space X with the Attouch–Wets topology. It is shown that the space Cld _AW (X) and its various subspaces are AR's. Moreover, if X is an infinite-dimensional Banach space with weight w(X) then Cld _AW (X) is homeomorphic to a Hilbert space with weight 2 ^w(X).     

A “planar” representation for generalized transition kernels

In [9], Mauldin, Preiss and von Weizsäcker have given a theorem representing transition kernels (atomless and between standard Borel spaces) by a planar model. Here, motivated by measure-theoretic as well as probabilistic considerations, we generalize by allowing the parametrizing spaceX to be arbitrary, with an arbitrary σ-field of “Borel” subsets, and allowing the corresponding measures to have atoms. (We also, for convenience rather than generality, allow arbitrary finite measures rather than probability ones.) The transition kernel is replaced by a substantially equivalent one fromX toX ×I that is “sectioned”, hence completely orthogonal. This is shown to be isomorphic to a model in which the image space consists of 3 specifically defined subsets ofX × ?: an ordinate set (in which vertical sections have Lebesgue measure), an “atomic” set contained inX × (??), and a “singular” set with null sections. The method incidentally produces and exploits a “reverse” transition kernel fromX toX ×I. Some further extensions are briefly discussed; in particular, allowing “uniformly σ-finite” measures (in the “standard” case) leads to a generalization that includes the planar representation theorem of Rokhlin [10] and the author [5]; cf. also [7, 2].     

On scatteredly continuous maps between topological spaces

A map f:X→Y between topological spaces is defined to be scatteredly continuous if for each subspace AX the restriction f|A has a point of continuity. We show that for a function f:X→Y from a perfectly paracompact hereditarily Baire Preiss–Simon space X into a regular space Y the scattered continuity of f is equivalent to (i) the weak discontinuity (for each subset AX the set D(f|A) of discontinuity points of f|A is nowhere dense in A), (ii) the piecewise continuity (X can be written as a countable union of closed subsets on which f is continuous), (iii) the G_δ-measurability (the preimage of each open set is of type G_δ). Also under Martin Axiom, we construct a G_δ-measurable map f:X→Y between metrizable separable spaces, which is not piecewise continuous. This answers an old question of V. Vinokurov.     

Iterations of quasi-uniform hyperspace constructions

Summary Given a quasi-uniform space (X,U), we study its Hausdorff quasi-uniformity U_H on the set P₀(X) of nonempty subsets of the set X. In particular we are concerned with the question whether at a certain finite stage iterations of the described Hausdorff hyperspace construction applied to two distinct quasi-uniformities on X will necessarily lead to hyperspaces carrying distinct induced topologies.