J. Chaber

On subparacompactness and related properties

Abstract An approach to the theory of subparacompactness is presented here. This approach allows one to understand the notions of subexpandability and to generalize a theorem from [6]. We also give an answer to a question from [5].

Mappings onto metric spaces

Abstract We give characterizations of perfect images and open and compact images of spaces that can be mapped onto metrizable spaces by a mapping with fibers having a given property P . We use these characterizations to obtain conditions which imply that such images can be mapped onto a metric space by a mapping with fibers satisfying P . Such a treatment includes the investigation of spaces with a weaker metric topology [2, Ch. 5].

On Souslin sets and embeddings in integer-valued function spaces on ω1

Abstract Let Σ( N ℵ 1 ) be the subspace of the t%1-product of natural numbers N ℵ 1 , consisting of functions with countable support. We prove that for any uncountable Souslin set A in gS( N ℵ 1 ) , either A contains a Cantor set, or a copy of ω1 (the space of countable ordinals) or else A can be well-ordered in type ω1 so that all initial segments are closed (Theorem 1.1). We give also a more refined version of this result (Theorem 1.2). In particular, we demonstrate non-effectiveness of some selections from natural “layers” in Σ( N ℵ 1 ) , extending some ideas of A.H. Stone concerning Borel theory in nonseparable metrizable spaces. Connections of this subject to classical Lusins constituents are also discussed. In another direction, we indicate (Corollary 1.3) a locally countable non-Souslin set in N ℵ 1 (witnessing poor covering properties of N ℵ 1 and answering a question by Kemoto and Yajima), and we find a closed perfectly normal subspace of N ℵ1 which is not a countable union of closed subsets with finite covering dimension.

On θ-refinability of strict p-spaces

Abstract In this paper, some partial solutions are given to the problem concerning θ -refinability of strict p -spaces. It is shown that every locally compact strict p -space is θ -refinable and every locally hereditarily separable strict p -space is subparacompact.

Closed mappings onto metric spaces

Abstract We investigate the classes of spaces that can be mapped onto a metrizable space by a closed mapping with fibers having a given property P . We give some conditions which assure that such classes are closed under the action of perfect or open and compact mappings. Such a treatment includes the investigation of paracompact p-spaces and M-spaces. We also discuss spaces that can be mapped onto a metacompact Moore space.

Open finite-to-one images of metric spaces

Abstract We give a characterization of open finite-to-one images of metric spaces and apply this characterization in the investigation of open finite-to-one images of paracompact p -spaces.

Two subclasses of the class MOBI

Abstract Assuming that all spaces are regular we prove that open and compact images of σ-locally compact metric spaces are σ-locally compact metacompact Moore spaces while open and compact images of σ-locally compact metacompact Moore spaces form the class of spaces with a point-countable base of countable order and a closure-preserving closed cover by σ-compact sets. Moreover, this class is the minimal class of regular spaces which contains all σ-locally compact metric spaces and is invariant under open and compact mappings. The complete version of these results gives a characterization of images of C -scattered metric spaces.