For a compact Hausdorff space X, let J be the ordered set associated with of all finite open covers X such that there exists nJ, where nJ is dimension ∂. Therefore, we have Hˇp(X;Z), 0≤p≤n=nJ. continuous self-map f on α∈J an cover and Lf(α)={Lf(U)|U∈α}. Then, fiber L˙f(α) Xf induced by Lf(α). In this paper, define topological entropy entL(f) as supremum ent(f,L˙f(α)) through Xf={Lf(U);U⊂X}, Lf(...

We give a general version of Bryc’s theorem valid on any topological space and with any algebra A of real-valued continuous functions separating the points, or any wellseparating class. In absence of exponential tightness, and when the underlying space is locally compact regular and A constituted by functions vanishing at infinity, we give a sufficient condition on the functional Λ(·)|A to get ...

We define and study a quantale-valued Wijsman structure on the hyperspace of all non-empty closed sets of a quantale-valued metric space. We show its admissibility and that the metrical coreflection coincides with the quantale-valued Hausdorff metric and that, for a metric space, the topological coreflection coincides with the classical Wijsman topology. We further define an index of compactnes...

We give two proofs of a representation theorem for those totally finite signed Radon measures on an infinite power K of a compact Hausdorff space K (with its product topology) with the property of invariance under permutation of coordinates, in terms of powers of Radon probability measures on K. This is essentially a version of de Finetti’s theorem on exchangeable sequences of random variables....

In this paper we introduce and study so-called k∗-metrizable spaces forming a new class of generalized metric spaces, and display various applications of such spaces in topological algebra, functional analysis, and measure theory. By definition, a Hausdorff topological space X is k∗-metrizable if X is the image of a metrizable space M under a continuous map f : M → X having a section s : X → M ...

We consider an infinite dimensional separable Hilbert space and its family of compact integrable cocycles over a dynamical system f . Assuming that f acts in a compact Hausdorff space X and preserves a Borel regular ergodic measure which is positive on non-empty open sets, we conclude that there is a residual subset of cocycles within which, for almost every x, either the OseledetsRuelle’s deco...

We give some necessary and sufficient conditions for the character of spaces to be preserved under closed maps. Introduction. Recall the following result due to K. Morita and S. Hanai [11] or A. H. Stone [14]: Let Y be the image of a metric space X under a closed map/. Then Y is first countable (or metrizable) if and only if every boundary df~x(y) of the point-inverse/"'^) is compact. E. Michae...

We prove that the maximal order type of the wqo of linear orders of finite Hausdorff rank under embeddability is φ2(0), the first fixed point of the ε-function. We then show that Fräıssé’s conjecture restricted to linear orders of finite Hausdorff rank is provable in ACA0 + “φ2(0) is well-ordered” and, over RCA0, implies ACA ′ 0 + “φ2(0) is well-ordered”.

Let G be a non-empty closed (resp. bounded closed) boundedly relatively weakly compact subset in a strictly convex Kadec Banach space X. Let K(X) denote the space of all non-empty compact convex subsets of X endowed with the Hausdorff distance. Moreover, let KG(X) denote the closure of the set {A ∈ K(X) : A∩G = ∅}. We prove that the set of all A ∈ KG(X) (resp. A ∈ K(X)), such that the minimizat...

We prove a large deviation principle for Minkowski sums of i.i.d. random compact sets in a Banach space, that is, the analog of Cramér theorem for random compact sets. Several works have been devoted to deriving limit theorems for random sets. For i.i.d. random compact sets in R, the law of large numbers was initially proved by Artstein and Vitale [1] and the central limit theorem by Cressie [3...