Selections and hyperspaces of finite sets

It is demonstrated that the hyperspace of at most (n+1)-point sets has a Vietoris continuous selection if both the hyperspace of at most n-point sets and that of exactly (n+1)-point sets have Vietoris continuous selections. This result is applied to demonstrate that the hyperspace of at most (2n+2)-point sets has a Vietoris continuous selection provided that one of at most (2n+1)-point sets has such a selection. This settles some open questions.     

Ordinals and set-valued zero-selections for hyperspaces

A continuous zero-selection f for the Vietoris hyperspace F(X) of the nonempty closed subsets of a space X is a Vietoris continuous map f:F(X)→X which assigns to every nonempty closed subset an isolated point of it. It is well known that a compact space X has a continuous zero-selection if and only if it is an ordinal space, or, equivalently, if X can be mapped onto an ordinal space by a continuous one-to-one surjection. In this paper, we prove that a compact space X has an upper semi-continuous set-valued zero-selection for its Vietoris hyperspace F(X) if and only if X can be mapped onto an ordinal space by a continuous finite-to-one surjection.     

The Lindel?f number of Cp(X)×Cp(X) for strongly zero-dimensional X

We prove that if X is a strongly zero-dimensional space, then for every locally compact second-countable space M, C _p (X, M) is a continuous image of a closed subspace of C _p (X). It follows in particular, that for strongly zero-dimensional spaces X, the Lindel?f number of C _p (X)×C _p (X) coincides with the Lindel?f number of C _p (X). We also prove that l(C _p (X ⁿ )^κ) ≤ l(C _p (X)^κ) whenever κ is an infinite cardinal and X is a strongly zero-dimensional union of at most κcompact subspaces.     

On c-realcompact spaces

Abstract

The c-realcompact spaces are fully studied and most of the important and well-known properties of realcompact spaces are extended to these spaces. For a zero-dimensional space X, the space υ₀X, which is the counterpart of υX, the Hewitt realcompactification of X, is introduced and studied. It is shown that υ₀X, which is the smallest c-realcompact space between X and β₀X, plays the same role (with respect to C_c(X)) as υX does in the context of C(X). It is proved for strongly zero-dimensional spaces, c-realcompact spaces, realcompact spaces and N-compact spaces coincide. In particular, if X is a strongly zero-dimensional space, then υX = υ₀X. It is obsesrved that a zero-dimensional space X is pseudocompact if and only if C_c(X) = C*_c(X), or equivalently if and only if υ₀X = β₀ X. In particular, a zero-dimensional pseudocompact space is compact if and only if it is c-realcompact. It is shown that Lindelöf spaces, subspaces of the one-point compactification (resp., Lindelöffication) of a discrete space with a nonmeasurable cardinal, are c-realcompact space. If X is a pseudocompact space, it is observed that C(X) = C_c(X) if and only if βX is scattered. Finally, the simplest possible proof (with reasoning) among the known proofs, of the well-known fact that discrete spaces of cardinality less than or equal to that of the continuum are realcompact, is given.     

Function Spaces and Dieudonné Completeness

Abstract

For a completely regular space X and a normed space E let C_k (x, E) (resp., C_p (x, E)) be the set of all E-valued continuous maps on X endowed with the compact-open (resp., pointwise convergence) topology. It is shown that the set of all F-valued linear continuous maps on C_k (x, E) when equipped with the topology of uniform convergence on the members of some families of bounded subsets of C_k (x, E) is a complete uniform space if F is a Band space and X is Dieudonné complete. This result is applied to prove that Dieudonné completeness is preserved by linear quotient surjections from C_k (x, E) onto C_k (Y, E) (resp., from C_p (x, E) onto C_p (x, E)) provided E, F are Band spaces and Y is a k-space.     

A metrizable X with Cp(X) not homeomorphic to Cp(X) × Cp(X)

We give an example of an infinite metrizable space X such that the space C_p(X), of continuous real-valued functions on X endowed with the pointwise topology, is not homeomorphic to its own square C_p(X) × C_p(X). The space X is a zero-dimensional subspace of the real line. Our result answers a long-standing open question in the theory of function spaces posed by A. V. Arhangel’skii.     

On spaces of p-adic vector measures

LetX be a Hausdorff zero-dimensional topological space,K(X) the algebra of all clopen subsets of X, E a Hausdorff locally convex space over a non-Archimedean valued field and C _b (X) the space of all bounded continuous -valued functions on X. The space M(K(X),E), of all bounded finitely-additive measures m: K(X) → E, is investigated. If we equip C _b (X) with the topologies β _o , β, β _u , τ _b or β _ob , it is shown that, for E (compete, the corresponding spaces of continuous linear operators from C _b (X) to E (are algebraically isomorphic to certain subspaces of M(K(X),E). The text was submitted by the author in English.     

Variations on a theorem of Arhangel’skii and Pytkeev

We identify continuous real-valued functions on a Tychonoff space X with their (closed) graphs thus allowing for C(X) to naturally inherit the lower Vietoris topology from the ambient hyperspace. We then calculate a bitopological version of tightness using the weak Lindelöf numbers of finite powers of X. We also characterize bitopological versions of countable fan and strong fan tightness of the point-open topology with respect to the lower Vietoris topology on C(X) in terms of suitable covering properties of the powers X ⁿ formulated using the language of S ₁ and S _fin selection principles.     

Cosmic spaces which are not μ-spaces among function spaces with the topology of pointwise convergence

A space is called a μ-space if it can be embedded in a countable product of paracompact F_σ-metrizable spaces. The following are shown:(1) For a Tychonoff space X, if C_p(X,R) is a μ-space, then X is a countable union of compact metrizable subspaces.(2) For a zero-dimensional space X, C_p(X,2) is a μ-space if and only if X is a countable union of compact metrizable subspaces.In particular, let P be the space of irrational numbers. Then C_p(P,2) is a cosmic space (i.e., a space with a countable network) which is not a μ-space.     

Vietoris topology on partial maps with compact domains

The space PK of partial maps with compact domains (identified with their graphs) forms a subspace of the hyperspace of nonempty compact subsets of a product space endowed with the Vietoris topology. Various completeness properties of PK, including ?ech-completeness, sieve completeness, strong Choquetness, and (hereditary) Baireness, are investigated. Some new results on the hyperspace K(X) of compact subsets of a Hausdorff X with the Vietoris topology are obtained; in particular, it is shown that there is a strongly Choquet X, with 1st category K(X).     

Metrizable compacta in the space of continuous functions with the topology of pointwise convergence

We prove that every point-finite family of nonempty functionally open sets in a topological space X has the cardinality at most an infinite cardinal κ if and only if w(X) ≦ κ for every Valdvia compact space Y C _p (X). Correspondingly a Valdivia compact space Y has the weight at most an infinite cardinal κ if and only if every point-finite family of nonempty open sets in C _p (Y) has the cardinality at most κ, that is p(C _p (Y)) ≦ κ. Besides, it was proved that w(Y) = p(C _p (Y)) for every linearly ordered compact Y. In particular, a Valdivia compact space or linearly ordered compact space Y is metrizable if and only if p(C _p (Y)) = ℵ₀. This gives answer to a question of O. Okunev and V. Tkachuk.      

A Nagata-like theorem for certain function spaces

Let X be a space, and let A be a zero-dimensional topological ring. In this paper we will consider a few natural questions that arise when studying the space C _p (X, A), the ring of continuous functions from X to A, endowed with the topology of pointwise convergence. It will be shown that the zero-dimensionality of the codomain plays a vital role in this study. An upper and lower bound will be determined for the density of C _p (X, A) using the density of A and the weight of X. The character of C _p (X, A) will be computed, thus characterizing when C _p (X, A) is metrizable. Lastly, we will consider the topological dual space of C _p (X, A) and use it to prove a Nagata-like theorem.     

Approximations of relations by continuous functions

Let X be a Tychonoff space, C(X) be the space of all continuous real-valued functions defined on X and CL(X×R) be the hyperspace of all nonempty closed subsets of X×R. We prove the following result. Let X be a countably paracompact normal space. The following are equivalent: (a) dimX=0; (b) the closure of C(X) in CL(X×R) with the Vietoris topology consists of all F∈CL(X×R) such that F(x)≠∅ for every x∈X and F maps isolated points into singletons; (c) each usco map which maps isolated points into singletons can be approximated by continuous functions in CL(X×R) with the locally finite topology. From the mentioned result we can also obtain the answer to Problem 5.5 in [L'. Holá, R.A. McCoy, Relations approximated by continuous functions, Proc. Amer. Math. Soc. 133 (2005) 2173-2182] and to Question 5.5 in [R.A. McCoy, Comparison of hyperspace and function space topologies, Quad. Mat. 3 (1998) 243-258] in the realm of normal, countably paracompact, strongly zero-dimensional spaces. Generalizations of some results from [L'. Holá, R.A. McCoy, Relations approximated by continuous functions, Proc. Amer. Math. Soc. 133 (2005) 2173-2182] are also given.     

A stability property for locally uniformly rotund renorming

It is proved that C(K,E) (the space of all continuous functions on a Hausdorff compact space K taking values in a Banach space E) admits an equivalent locally uniformly rotund norm if C(K) and E do so. Moreover, if the equivalent LUR norms on C(K) and E are lower semicontinuous with respect to some weak topologies, the LUR norm on C(K,E) can be chosen to be lower semicontinuous with respect to an appropriate weak topology. As a consequence we prove that if X and Y are two Hausdorff compacta and C(X), C(Y) admit equivalent (pointwise lower semicontinuous) LUR norms, then so does C(X×Y).     

A unified approach to theories of shadowing

This paper introduces the notion of a general approximation property, which encompasses many existing types of shadowing. It is proven that there exists a metric space X such that the sets of maps with many types of general approximation properties (including the classic shadowing, the L _p -shadowing, limit shadowing, and the s-limit shadowing) are not dense in C(X), S(X), and H(X) (the space of continuous self-maps of X, continuous surjections of X onto itself, and self-homeomorphisms of X) and that there exists a manifold M such that the sets of maps with general approximation properties of nonlocal type (including the average shadowing property and the asymptotic average shadowing property) are not dense in C(M), S(M), and H(M). Furthermore, it is proven that the sets of maps with a wide range of general approximation properties (including the classic shadowing, the L _p -shadowing, and the s-limit shadowing) are dense in the space of continuous self-maps of the Cantor set. A condition is given that guarantees transfer of general approximation property from a map on X to the map induced by it on the hyperspace of X. It is also proven that the transfer in the opposite direction always takes place.     

On (strong) α-favorability of the Vietoris hyperspace

For a normal space X, α (i.e. the nonempty player) having a winning strategy (resp. winning tactic) in the strong Choquet game Ch(X) played on X is equivalent to α having a winning strategy (resp. winning tactic) in the strong Choquet game played on the hyperspace CL(X) of nonempty closed subsets endowed with the Vietoris topology τ _V . It is shown that for a non-normal X where α has a winning strategy (resp. winning tactic) in Ch(X), α may or may not have a winning strategy (resp. winning tactic) in the strong Choquet game played on the Vietoris hyperspace. If X is quasi-regular, then having a winning strategy (resp. winning tactic) for α in the Banach-Mazur game BM(X) played on X is sufficient for α having a winning strategy (resp. winning tactic) in BM(CL(X), τ _V ), but not necessary, not even for a separable metric X. In the absence of quasi-regularity of a space X where α has a winning strategy in BM(X), α may or may not have a winning strategy in the Banach-Mazur game played on the Vietoris hyperspace.     

The Choquet integral representation in the affine vector-valued case

LetX be any compact convex subset of a locally convex Hausdorff space andE be a complex Banach space. We denote byA(X, E) the space of all continuous and affineE-valued functions defined onX. In this paper we prove thatX is a Choquet simplex if and only if the dual ofA(X, E) is isometrically isomorphic by a selection map toM _m (X, E*), the space ofE*-valued,w*-regular boundary measures onX. This extends and strengthens a result of G. M. Ustinov. To do this we show that for any compact convex setX, each element of the dual ofA(X, E) can be represented by a measure inM _m (X, E*) with the same norm, and this representation is unique if and only ifX is a Choquet simplex. We also prove that ifX is metrizable andE is separable then there exists a selection map from the unit ball of the dual ofA(X, E) into the unit ball ofM _m (X, E*) which is weak* to weak*-Borel measurable.This work will constitute a portion of the author's Ph.D. Thesis at the University of Illinois.     

Coincidence of the Isbell and fine Isbell topologies

Let C(X,Y) be the set of all continuous functions from a topological space X into a topological space Y. We find conditions on X that make the Isbell and fine Isbell topologies on C(X,Y) equal for all Y. For zero-dimensional spaces X, we show there is a space Z such that the coincidence of the Isbell and fine Isbell topologies on C(X,Z) implies the coincidence on C(X,Y) for all Y. We then consider the question of when the Isbell and fine Isbell topologies coincide on the set of continuous real-valued functions. Our results are similar to results established for consonant spaces.     

p-Lattice Summing Operators

In this paper we study the spaces ∞_p(E, X) of p-lattice summing operators from a Banach space E to a Banach lattice X. The main results characterize those E and X for which Δ_p(E, X) = II_p(E, X) and we show that ∞(E, X)=Δ₂(E, X) for an infinite dimensional Banach lattice X of finite cotype if and only if E is isomorphic to a Hilbert space.     

Wijsman and Hit-and-Miss Topologies of Quasi-Metric Spaces

The relationship between the Wijsman topology and (proximal) hit-and-miss topologies is studied in the realm of quasi-metric spaces. We establish the equivalence between these hypertopologies in terms of Urysohn families of sets. Our results generalize well-known theorems and provide easier proofs. In particular, we prove that for a quasi-pseudo-metrizable space (X,T) the Vietoris topology on the set P ₀(X) of all nonempty subsets of X is the supremum of all Wijsman topologies associated with quasi-pseudo-metrics compatible with T. We also show that for a quasi-pseudo-metric space (X,d) the Hausdorff extended quasi-pseudo-metric is compatible with the Wijsman topology on P ₀(X) if and only if d ^–1 is hereditarily precompact.