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.     

The D-property of some Lindelöf spaces and related conclusions

It is shown that the space Cp(τω) is a D-space for any ordinal number τ, where . This conclusion gives a positive answer to R.Z. Buzyakova's question. We also prove that another special example of Lindelöf space is a D-space. We discuss the D-property of spaces with point-countable weak bases. We prove that if a space X has a point-countable weak base, then X is a D-space. By this conclusion and one of T. Hoshina's conclusion, we have that if X is a countably compact space with a point-countable weak base, then X is a compact metrizable space. In the last part, we show that if a space X is a finite union of θ-refinable spaces, then X is a αD-space.     

On linear neighborhood assignments and dually discrete spaces

In this note, the concept of a linear neighborhood assignment is introduced. By discussing properties of linear D-spaces, we show that if T is a Suslin tree with FW (or CW) topology, then T is a Lindelöf D-space. We also show that if X is a countably compact space and , where for any linear neighborhood assignment ?n for Xn, there exists a strong DC-like subspace (or a subparacompact C-scattered closed subspace) Dn of Xn, such that for each n∈N, then X is a compact space; Every generalized ordered space is dually discrete. This gives a positive answer to a question of Buzyakova, Tkachuk and Wilson.     

On continuous choice of retractions onto nonconvex subsets

For a Banach space B and for a class A of its bounded closed retracts, endowed with the Hausdorff metric, we prove that retractions on elements A∈A can be chosen to depend continuously on A, whenever nonconvexity of each A∈A is less than . The key geometric argument is that the set of all uniform retractions onto an α-paraconvex set (in the spirit of E. Michael) is -paraconvex subset in the space of continuous mappings of B into itself. For a Hilbert space H the estimate can be improved to and the constant can be replaced by the root of the equation α+α²+α³=1.     

Preserving Z-sets by Dranishnikov's resolution

We prove that Dranishnikov's k-dimensional resolution is a UV^n − 1-divider of Chigogidze's k-dimensional resolution ck. This fact implies that preserves Z-sets. A further development of the concept of UV^n − 1-dividers permits us to find sufficient conditions for to be homeomorphic to the Nöbeling space νk or the universal pseudoboundary σk. We also obtain some other applications.     

On the topological Helly theorem

The main result of this paper is the following theorem, related to the missing link in the proof of the topological version of the classical result of Helly: Let be any family of simply connected compact subsets of R² such that for every i,j∈{0,1,2} the intersections Xi∩Xj are path connected and is nonempty. Then for every two points in the intersection there exists a cell-like compactum connecting these two points, in particular the intersection is a connected set.     

On some classes of Lindelöf Σ-spaces

We consider special subclasses of the class of Lindelöf Σ-spaces obtained by imposing restrictions on the weight of the elements of compact covers that admit countable networks: A space X is in the class LΣ(?κ) if it admits a cover by compact subspaces of weight κ and a countable network for the cover. We restrict our attention to κ?ω. In the case κ=ω, the class includes the class of metrizably fibered spaces considered by Tkachuk, and the P-approximable spaces considered by Tka?enko. The case κ=1 corresponds to the spaces of countable network weight, but even the case κ=2 gives rise to a nontrivial class of spaces. The relation of known classes of compact spaces to these classes is considered. It is shown that not every Corson compact of weight ℵ₁ is in the class LΣ(?ω), answering a question of Tkachuk. As well, we study whether certain compact spaces in LΣ(?ω) have dense metrizable subspaces, partially answering a question of Tka?enko. Other interesting results and examples are obtained, and we conclude the paper with a number of open questions.     

The Namioka property of KC-functions and Kempisty spaces

A topological space Y is called a Kempisty space if for any Baire space X every function , which is quasi-continuous in the first variable and continuous in the second variable has the Namioka property. Properties of compact Kempisty spaces are studied in this paper. In particular, it is shown that any Valdivia compact is a Kempisty space and the Cartesian product of an arbitrary family of compact Kempisty spaces is a Kempisty space.     

On a question by Alexey Ostrovsky concerning preservation of completeness

Let be a surjection of a zero-dimensional metrizable X onto a metrizable Y which maps clopen sets in X to locally closed (or more generally, resolvable) sets in Y. We prove that if X is completely metrizable, or hereditarily Baire, then Y has also the respective property. This strengthens some recent results of A. Ostrovsky (2009) [5] and provides an answer to his question.     

Certain sequences with compact closure

This paper deals with a question which is stated by quite simple definitions. A sequence {xn} in a space X is called a β-sequence if every subsequence of it has a cluster point in X. The closure of the sequence {xn} means the closure of in X. Here we consider the question when a β-sequence has compact closure. We give several answers to this question.     

A glance at spaces with closure-preserving local bases

Call a space X (weakly) Japanese at a pointx∈X if X has a closure-preserving local base (or quasi-base respectively) at the point x. The space X is (weakly) Japanese if it is (weakly) Japanese at every x∈X. We prove, in particular, that any generalized ordered space is Japanese and that the property of being (weakly) Japanese is preserved by σ-products; besides, a dyadic compact space is weakly Japanese if and only if it is metrizable. It turns out that every scattered Corson compact space is Japanese while there exist even Eberlein compact spaces which are not weakly Japanese. We show that a continuous image of a compact first countable space can fail to be weakly Japanese so the (weak) Japanese property is not preserved by perfect maps. Another interesting property of Japanese spaces is their tightness-monolithity, i.e., in every weakly Japanese space X we have for any set A⊂X.     

Atsuji completions: Equivalent characterisations

A metric space (X,d) is called an Atsuji space if every real-valued continuous function on (X,d) is uniformly continuous. It is well known that an Atsuji space must be complete. A metric space (X,d) is said to have an Atsuji completion if its completion is an Atsuji space. In this paper, we study twenty-nine equivalent characterisations for a metric space to have an Atsuji completion.