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.     

More on perfectly normal non-realcompact spaces

We use the space associated with a guessing sequence on ω₁ to show that it is consistent with CH that there exists a locally countable, first-countable, locally compact, perfectly normal, non-realcompact space of size ℵ₁ which does not contain any sub-Ostaszewski spaces. By a similar technique, it is shown to be consistent with that there exists a locally countable, first-countable, perfectly normal, non-realcompact space of size ℵ₁.     

On spaces which are D, linearly D and transitively D

In this note, we comment on D-spaces, linearly D-spaces and transitively D-spaces. We show that every meta-Lindelöf space is transitively D. If X is a weak -refinable TD-scattered space, then X is transitively D, where TD is the class of all transitively D-spaces. If X is a weak -refinable -scattered space, then X is a D-space, where is the class of all D-spaces, and hence every weak -refinable (or submetacompact) scattered space is a D-space. This gives a positive answer to a question mentioned by Martínez and Soukup. In the last part of this note, we show that if X is a weak -refinable space then X is linearly D.     

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.     

Distances from selectors to spaces of Baire one functions

Given a metric space X and a Banach space (E,‖⋅‖) we study distances from the set of selectors Sel(F) of a set-valued map to the space B₁(X,E) of Baire one functions from X into E. For this we introduce the d-τ-semioscillation of a set-valued map with values in a topological space (Y,τ) also endowed with a metric d. Being more precise we obtain that

     

On a question concerning sharp bases

A sharp base B is a base such that whenever (Bi)_i<ω is an injective sequence from B with x∈_?i<ωBi, then is a base at x. Alleche, Arhangel'ski? and Calbrix asked: if X has a sharp base, must X×[0,1] have a sharp base? Good, Knight and Mohamad claimed to construct an example of a Tychonoff space P with a sharp base such that P×[0,1] does not have a sharp base. However, the space was not regular. We show how to modify the construction to make P Tychonoff.     

Minimal pseudocompact group topologies on free abelian groups

A Hausdorff topological group G is minimal if every continuous isomorphism f:G→H between G and a Hausdorff topological group H is open. Significantly strengthening a 1981 result of Stoyanov, we prove the following theorem: For every infinite minimal abelian group G there exists a sequence of cardinals such that

     

Several remarks on dimensions modulo ANR-compacta

We investigate a dimension function L-dim (L is a class of ANR-compacta). Main results are as follows.Let L be an ANR-compactum.(1) If L*L is not contractible, then for every n?0 there is a cube Im with .(2) If L is simply connected and f:X→Y is an acyclic mapping from a finite-dimensional compact Hausdorff space X onto a finite-dimensional space Y, then .(3) If L is simply connected and L*L is not contractible, then for every n?2 there exists a compact Hausdorff space such that , and for an arbitrary closed set either or .     

Dually discrete spaces

A neighbourhood assignment in a space X is a family of open subsets of X such that x∈Ox for any x∈X. A set Y⊆X is a kernel ofO if . We obtain some new results concerning dually discrete spaces, being those spaces for which every neighbourhood assignment has a discrete kernel. This is a strictly larger class than the class of D-spaces of [E.K. van Douwen, W.F. Pfeffer, Some properties of the Sorgenfrey line and related spaces, Pacific J. Math. 81 (2) (1979) 371-377].