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).     

A non-separable Christensen's theorem and set tri-quotient maps

For every space X let K(X) be the set of all compact subsets of X. Christensen [J.P.R. Christensen, Necessary and sufficient conditions for measurability of certain sets of closed subsets, Math. Ann. 200 (1973) 189-193] proved that if X,Y are separable metrizable spaces and F:K(X)→K(Y) is a monotone map such that any L∈K(Y) is covered by F(K) for some K∈K(X), then Y is complete provided X is complete. It is well known [J. Baars, J. de Groot, J. Pelant, Function spaces of completely metrizable space, Trans. Amer. Math. Soc. 340 (1993) 871-879] that this result is not true for non-separable spaces. In this paper we discuss some additional properties of F which guarantee the validity of Christensen's result for more general spaces.     

Linearly ordered compacta and Banach spaces with a projectional resolution of the identity

We construct a compact linearly ordered space Kω₁ of weight ℵ₁, such that the space C(Kω₁) is not isomorphic to a Banach space with a projectional resolution of the identity, while on the other hand, Kω₁ is a continuous image of a Valdivia compact and every separable subspace of C(Kω₁) is contained in a 1-complemented separable subspace. This answers two questions due to O. Kalenda and V. Montesinos.     

Weak topology and Browder-Kirk's theorem on hyperspace

Let K be a weakly compact, convex subset of a Banach space X with normal structure. Browder-Kirk's theorem states that every non-expansive mapping T which maps K into K has a fixed point in K. Suppose now that WCC(X) is the collection of all non-empty weakly compact convex subsets of X. We shall define a certain weak topology Tw on WCC(X) and have the above-mentioned result extended to the hyperspace (WCC(X);Tw).     

Chain conditions and weak topologies

We study conditions on Banach spaces close to separability. We say that a topological space is pcc if every point-finite family of open subsets of the space is countable. For a Banach space E, we say that E is weakly pcc if E, equipped with the weak topology, is pcc, and we also consider a weaker property: we say that E is half-pcc if every point-finite family consisting of half-spaces of E is countable. We show that E is half-pcc if, and only if, every bounded linear map E→c₀(ω₁) has separable range. We exhibit a variety of mild conditions which imply separability of a half-pcc Banach space. For a Banach space C(K), we also consider the pcc-property of the topology of pointwise convergence, and we note that the space Cp(K) may be pcc even when C(K) fails to be weakly pcc. We note that this does not happen when K is scattered, and we provide the following example:

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There exists a non-metrizable scattered compact Hausdorff space K with C(K) weakly pcc.

     

Generic uniqueness of equilibrium points

Let X be a nonempty, convex and compact subset of normed linear space E (respectively, let X be a nonempty, bounded, closed and convex subset of Banach space E and A be a nonempty, convex and compact subset of X) and f:X×X→R be a given function, the uniqueness of equilibrium point for equilibrium problem which is to find x^∗∈X (respectively, x^∗∈A) such that f(x^∗,y)≥0 for all y∈X (respectively, f(x^∗,y)≥0 for all y∈A) is studied with varying f (respectively, with both varying f and varying A). The results show that most of equilibrium problems (in the sense of Baire category) have unique equilibrium point.     

Submaximal and door compactifications

In this paper, a characterization is given for compact door spaces. We, also, deal with spaces X such that a compactification K(X) of X is submaximal or door.Let X be a topological space and K(X) be a compactification of X.We prove, here, that K(X) is submaximal if and only if for each dense subset D of X, the following properties hold:

(i)

D is co-finite in K(X);

(ii)

for each x∈K(X)?D, {x} is closed.

If X is a noncompact space, then we show that K(X) is a door space if and only if X is a discrete space and K(X) is the one-point compactification of X.     

On one-point metrizable extensions of locally compact metrizable spaces

For a non-compact metrizable space X, let E(X) be the set of all one-point metrizable extensions of X, and when X is locally compact, let EK(X) denote the set of all locally compact elements of E(X) and be the order-anti-isomorphism (onto its image) defined in [M. Henriksen, L. Janos, R.G. Woods, Properties of one-point completions of a non-compact metrizable space, Comment. Math. Univ. Carolin. 46 (2005) 105-123; in short HJW]. By definition λ(Y)=_?n<ωclβX(Un∩X)\X, where Y=X∪{p}∈E(X) and {Un}_n<ω is an open base at p in Y. We characterize the elements of the image of λ as exactly those non-empty zero-sets of βX which miss X, and the elements of the image of EK(X) under λ, as those which are moreover clopen in βX\X. This answers a question of [HJW]. We then study the relation between E(X) and EK(X) and their order structures, and introduce a subset ES(X) of E(X). We conclude with some theorems on the cardinality of the sets E(X) and EK(X), and some open questions.     

Group-valued continuous functions with the topology of pointwise convergence

Let G be a topological group with the identity element e. Given a space X, we denote by Cp(X,G) the group of all continuous functions from X to G endowed with the topology of pointwise convergence, and we say that X is: (a) G-regular if, for each closed set F⊆X and every point x∈X?F, there exist f∈Cp(X,G) and g∈G?{e} such that f(x)=g and f(F)⊆{e}; (b) G^?-regular provided that there exists g∈G?{e} such that, for each closed set F⊆X and every point x∈X?F, one can find f∈Cp(X,G) with f(x)=g and f(F)⊆{e}. Spaces X and Y are G-equivalent provided that the topological groups Cp(X,G) and Cp(Y,G) are topologically isomorphic.We investigate which topological properties are preserved by G-equivalence, with a special emphasis being placed on characterizing topological properties of X in terms of those of Cp(X,G). Since R-equivalence coincides with l-equivalence, this line of research “includes” major topics of the classical Cp-theory of Arhangel'ski? as a particular case (when G=R).We introduce a new class of TAP groups that contains all groups having no small subgroups (NSS groups). We prove that: (i) for a given NSS group G, a G-regular space X is pseudocompact if and only if Cp(X,G) is TAP, and (ii) for a metrizable NSS group G, a G^?-regular space X is compact if and only if Cp(X,G) is a TAP group of countable tightness. In particular, a Tychonoff space X is pseudocompact (compact) if and only if Cp(X,R) is a TAP group (of countable tightness). Demonstrating the limits of the result in (i), we give an example of a precompact TAP group G and a G-regular countably compact space X such that Cp(X,G) is not TAP.We show that Tychonoff spaces X and Y are T-equivalent if and only if their free precompact Abelian groups are topologically isomorphic, where T stays for the quotient group R/Z. As a corollary, we obtain that T-equivalence implies G-equivalence for every Abelian precompact group G. We establish that T-equivalence preserves the following topological properties: compactness, pseudocompactness, σ-compactness, the property of being a Lindelöf Σ-space, the property of being a compact metrizable space, the (finite) number of connected components, connectedness, total disconnectedness. An example of R-equivalent (that is, l-equivalent) spaces that are not T-equivalent is constructed.     

Note on separate continuity and the Namioka property

A pair 〈B,K〉 is a Namioka pair if K is compact and for any separately continuous , there is a dense A⊆B such that f is ( jointly) continuous on A×K. We give an example of a Choquet space B and separately continuous such that the restriction fΔ_| to the diagonal does not have a dense set of continuity points. However, for K a compact fragmentable space we have: For any separately continuous and for any Baire subspace F of T×K, the set of points of continuity of is dense in F. We say that 〈B,K〉 is a weak-Namioka pair if K is compact and for any separately continuous and a closed subset F projecting irreducibly onto B, the set of points of continuity of fF_| is dense in F. We show that T is a Baire space if the pair 〈T,K〉 is a weak-Namioka pair for every compact K. Under (CH) there is an example of a space B such that 〈B,K〉 is a Namioka pair for every compact K but there is a countably compact C and a separately continuous which has no dense set of continuity points; in fact, f does not even have the Baire property.     

Domain representability of certain function spaces

Let Cp(X) be the space of all continuous real-valued functions on a space X, with the topology of pointwise convergence. In this paper we show that Cp(X) is not domain representable unless X is discrete for a class of spaces that includes all pseudo-radial spaces and all generalized ordered spaces. This is a first step toward our conjecture that if X is completely regular, then Cp(X) is domain representable if and only if X is discrete. In addition, we show that if X is completely regular and pseudonormal, then in the function space Cp(X), Oxtoby's pseudocompleteness, strong Choquet completeness, and weak Choquet completeness are all equivalent to the statement “every countable subset of X is closed”.     

A note on a theorem of Talagrand

We provide an elementary argument to show that if for a hemicompact kR-space X the space Cp(X) contains a subset S which separates the points of X and is dominated by irrationals, i.e. S is covered by a family of compact sets such that Kα⊂Kβ for α?β, then Cp(X) is also dominated by irrationals; consequently Cp(X) is K-analytic. This fact (which fails for non-hemicompact spaces X) extends an old result of Talagrand.     

Convex vector lattices and l-algebras

For A an Archimedean Riesz space (=vector lattice) with distinguished positive weak unit e_A, we have the Yosida representation  as a Riesz space in D(X_A), the lattice of extended real valued functions on the space of e_A-maximal ideas. This note is about those A for which  is a convex subset of D(X_A); we call such A “convex”.Convex Riesz spaces arise from the general issue of embedding as a Riesz ideal, from consideration of uniform- and order-completeness, and from some problems involving comparison of maximal ideal spaces (which we won't discuss here; see [10]).The main results here are: (2.4) A is convex iff A is contained as a Riesz ideal in a uniformly complete Φ-algebra B with identity e_A. (3.1) Any A has a convex reflection (i.e., embeds into a convex B with a universal mapping property for Riesz homomorphisms; moreover, the embedding is epic and large).     

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.     

The compact-open topology: A new perspective

This paper studies the compact-open topology on the set KC(X) of all real-valued functions defined on a Tychonoff space, which are continuous on compact subsets of X. In addition to metrizability, separability and second countability of this topology on KC(X), various kinds of topological properties of this topology are studied in detail. Actually the motivation for studying the compact-open topology on KC(X) lies in the attempt of having a simpler proof for the characterization of a completeness property of the compact-open topology on C(X), the set of all real-valued continuous functions on X.     

Linear extensions,projections, and split faces

The main result is essentially: Let F be a closed split face of a compact convex set K such that A(F) is separable and has the (positive) metric approximation property. Then there is a (positive) linear extension operator from A(F) into A(K) of norm one.This is applied to C¹-algebras thus giving sufficient conditions for the existence of right inverses to surjective ¹-homomorphisms.     

Domination by metric spaces

Following the definition of domination of a topological space X by a metric space M introduced by Cascales, Orihuela and Tkachuk (2011) in [3], we define a topological cardinal invariant called the metric domination index of a topological space X   as minimum of the set {w(M):M is a metric space that dominates X}$\{w(M):M\text{is a metric space that dominates}X\}$. This invariant quantifies or measures the concept of M-domination of Cascales et al. (2011) [3]. We prove (in ZFC) that if K   is a compact space such that _Cp(K)$C_p(K)$ is strongly dominated by a second countable space then K is countable. This answers a question by the authors of Cascales et al. (2011) [3].     

Pseudocompactness of hyperspaces

We consider the following question of Ginsburg: Is there any relationship between the pseudocompactness ofXωand that of the hyperspaceX²? We do that first in the context of Mrówka-Isbell spaces Ψ(A) associated with a maximal almost disjoint (MAD) family A on ω answering a question of J. Cao and T. Nogura. The space Ψω(A) is pseudocompact for every MAD family A. We show that

(1)

(p=c) ²Ψ(A) is pseudocompact for every MAD family A.

(2)

(h<c) There is a MAD family A such that ²Ψ(A) is not pseudocompact.

We also construct a ZFC example of a space X such that Xω is pseudocompact, yet X² is not.     

Function spaces over GO-spaces: Part I

We describe the structure of spaces of continuous step functions over GO-spaces. We establish a relation between the Dedekind completion of a GO-space L and properties of the space of continuous functions from L to 2 with finitely many steps. We use the established relation to prove that a countably compact GO-space L has Lindelöf Cp(L) iff the Dedekind remainder of L is Lindelöf and every compact subspace of L is metrizable. Or equivalently, a countably compact GO-space L has Lindelöf Cp(L) iff every compact subspace of L is metrizable and a Gδ-set in L. Other results are obtained.