Rudin's Dowker space, strong base-normality and base-strong-zero-dimensionality

It is well known that every pair of disjoint closed subsets F₀,F₁ of a normal T₁-space X admits a star-finite open cover U of X such that, for every U∈U, either or holds. We define a T₁-space X to be strongly base-normal if there is a base B for X with |B|=w(X) satisfying that every pair of disjoint closed subsets F₀,F₁ of X admits a star-finite cover B^′ of X by members of B such that, for every B∈B^′, either or holds. We prove that there is a base-normal space which is not strongly base-normal. Moreover, we show that Rudin's Dowker space is strongly base-(collectionwise)normal. Strong zero-dimensionality on base-normal spaces are also studied.     

The transitivity of induced maps

For a metric continuum X, we consider the hyperspaces X² and C(X) of the closed and nonempty subsets of X and of subcontinua of X, respectively, both with the Hausdorff metric. For a given map we investigate the transitivity of the induced maps and . Among other results, we show that if X is a dendrite or a continuum of type λ and is a map, then C(f) is not transitive. However, if X is the Hilbert cube, then there exists a transitive map such that f² and C(f) are transitive.     

Approximation of the Hausdorff distance by the distance of continuous surjections

The aim of this paper is to answer the following question: let (X,?) and (Y,d) be metric spaces, let A,B⊂Y be continuous images of the space X and let be a fixed continuous surjection. When is the inequality

     

Continuous selections and σ-spaces

Assume that X⊆R?Q, and each clopen-valued lower semicontinuous multivalued map has a continuous selection . Our main result is that in this case, X is a σ-space. We also derive a partial converse implication, and present a reformulation of the Scheepers Conjecture in the language of continuous selections.     

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.     

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.     

Exponentiable monomorphisms in categories of domains

We give a characterization of exponentiable monomorphisms in the categories of ω-complete posets, of directed complete posets and of continuous directed complete posets as those monotone maps f that are convex and that lift an element (and then a queue) of any directed set (ω-chain in the case of ) whose supremum is in the image of f (Theorem 1.9). Using this characterization, we obtain that a monomorphism f:X→B in (, ) exponentiable in w.r.t. the Scott topology is exponentiable also in (, ). We prove that the converse is true in the category , but neither in , nor in .     

Computation of blowing up centers

Given a birational projective morphism of quasi-projective varieties . We want to find the ideal sheaf over X such that the blowing up of X along corresponds to f. In this paper we approach the problem from two directions, solving two subcases. First we present a method that determines when f is the composition of blowing ups along known centers, then by another method, we compute directly from .     

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 .