Hyperspace Of Compact Subsets Research Articles

Symmetric products of Erdős space and complete Erdős space

It is shown that the symmetric products of complete Erdős space and Erdős space are homeomorphic to complete Erdős space and Erdős space, respectively. We will also give some properties of their hyperspace of compact subsets with the Vietoris topology.

On a hyperspace of compact subsets which is homeomorphic to a non-separable Hilbert space

Let X be a metrizable space and Comp(X) be the hyperspace consisting of non-empty compact subsets of X endowed with the Vietoris topology. In this paper, we give a necessary and sufficient condition on X for Comp(X) being homeomorphic to a non-separable Hilbert space. Moreover, we consider the topological structure of pair (Comp(X‾),Fin(X)) of hyperspaces of X and its completion X‾, where Fin(X) is the hyperspace of non-empty finite sets in X.

Formula omitted]-spaces

A regular topological space X is defined to be a P0-space if it has countable Pytkeev network. A network N for X is called a Pytkeev network if for any point x∈X, neighborhood Ox⊂X of x and subset A⊂X accumulating at a x there is a set N∈N such that N⊂Ox and N∩A is infinite. The class of P0-spaces contains all metrizable separable spaces and is (properly) contained in the Michael's class of ℵ0-spaces. It is closed under many topological operations: taking subspaces, countable Tychonoff products, small countable box-products, countable direct limits, hyperspaces of compact subsets. For an ℵ0-space X and a P0-space Y the function space Ck(X,Y) endowed with the compact-open topology is a P0-space. For any sequential ℵ0-space X the free abelian topological group A(X) and the free locally convex linear topological space L(X) both are P0-spaces. A sequential space is a P0-space if and only if it is an ℵ0-space. A topological space is metrizable and separable if and only if it is a P0-space with countable fan tightness.

Completeness of hyperspaces of compact subsets of quasi-metric spaces

We study the hyperspace K0(X) of non-empty compact subsets of a Smyth-complete quasi-metric space (X;d). We show that K0(X), equipped with the Hausdorfi quasi-pseudometric Hd forms a (sequentially) Yoneda-complete space. Moreover, if d is a T1 quasi-metric, then the hyperspace is algebraic, and the set of all flnite subsets forms a base for it. Finally, we prove that i K0(X);Hd ¢ is Smyth-complete if (X;d) is Smyth-complete and all compact subsets of X are d i1 -precompact.

The embedding of a mapping space with compact open topology

We give the embedding theorem of a mapping space C K ( X, Y) with compact open topology in C p ( K( X), K( Y)) with pointwise convergence topology, where K( X), K( Y) are the hyperspaces of compact subsets with finite topology.

On CF families and hyperspaces of compact subsets

First, we introduce the notion of CF families and study the relation with the other families. Second, we consider the properties of the hyperspace K X of nonempty compact subsets of a space X with a finite topology, when X is a Lašnev space or a space which satisfies some condition stated in terms of CF families, compact-finite families, and so on.

Completeness properties of hyperspaces of compact fuzzy sets

0. Introduction. On an arbitrary uniform space there are two types of compactlike fuzzy sets which are widely used in applications: u.s.c. fuzzy sets with compact support ( we denote this collection $C(X)) and u.s.c. fuzzy sets with compact levelsets (we denote this collection *iv(X)) [2], [12]. Always $C(X) C $w(X) but the converse holds only if X itself is compact. In the first part of our paper we prove that for the global fuzzy hyperspace structure [8], [9] the completeness of X is equivalent to the completeness of $ c (X) and to either the completeness or the ultracompleteness of $w (X) [6], [7]. In the second part we then prove the rather surprising result that the completion of $c(-^0 [7] is isomorphic to $jy (X) where X denotes the completion of X. These results not only generalize K. Morita's results on hyperspace of compact subsets [11] to the setting of fuzzy hyperspaces of compactlike fuzzy subsets but moreover via the isomorphism of the uniform modification of $C(X) and $w (X) with hyperspaces of closed subsets of X x [0,1] [9], they also include an extension of K. Morita's classical results to classes of closed subsets of X x [0,1] which are in general not compact.