Weak covering properties and the class MOBI
Weak-bases and -spaces
It is shown that certain weak-base structures on a topological space give a -space. This solves the question by A.V. Arhangel’skii of when quotient images of metric spaces are -spaces. A related result about symmetrizable spaces also answers a question of Arhangel’skii. Theorem.Any symmetrizable space is a -space hereditarily. Hence, quotient mappings, with compact fibers, from metric spaces have a -space image. What about quotient -mappings? Arhangel’skii and Buzyakova have shown that...
Weakly compactly generated Frechet spaces.
Weakly compatible maps in 2-non-Archimedean Menger PM-spaces.
Weakly complete semimetrizable spaces and complete metrizability.
In [4], J. Ceder proved that every paracompact strongly complete semimetrizable space is completely metrizable. This result cannot be generalized to paracompact weakly complete semimetrizable spaces as a known example of L. F. McAuley shows (see [11, Theorem 3.2]). It then arises, in a natural way, the question of obtaining conditions for the complete metrizability of a paracompact weakly complete semimetrizable space. In this note we give an answer to this question. We show that every regular theta,...
Weakly Contractive maps and Elementary Domain Invariance Theorem
Weak-open compact images of metric spaces
The main results of this paper are that (1) a space is -developable if and only if it is a weak-open image of a metric space, one consequence of the result being the correction of an error in the paper of Z. Li and S. Lin; (2) characterizations of weak-open compact images of metric spaces, which is another answer to a question in in the paper of Y. Ikeda, C. liu and Y. Tanaka.
Weil nearness spaces.
Weil uniformities for frames
In pointfree topology, the notion of uniformity in the form of a system of covers was introduced by J. Isbell in [11], and later developed by A. Pultr in [14] and [15]. Another equivalent notion of locale uniformity was given by P. Fletcher and W. Hunsaker in [6], which they called “entourage uniformity”. The purpose of this paper is to formulate and investigate an alternative definition of entourage uniformity which is more likely to the Weil pointed entourage uniformity, since it is expressed...
Wellchainedness characterizations of connected relators.
When are Borel functions Baire functions?
The following two theorems give the flavour of what will be proved. Theorem. Let Y be a complete metric space. Then the families of first Baire class functions and of first Borel class functions from [0,1] to Y coincide if and only if Y is connected and locally connected.Theorem. Let Y be a separable metric space. Then the families of second Baire class functions and of second Borel class functions from [0,1] to Y coincide if and only if for all finite sequences of nonempty open subsets of Y there...
When is domain representable
Let M be a metrizable group. Let G be a dense subgroup of . We prove that if G is domain representable, then . The following corollaries answer open questions. If X is completely regular and is domain representable, then X is discrete. If X is zero-dimensional, T₂, and is subcompact, then X is discrete.
When is any continuous function Lipschitzian?
When spectra of lattices of -ideals are Stone-Čech compactifications
Let be a completely regular Hausdorff space and, as usual, let denote the ring of real-valued continuous functions on . The lattice of -ideals of has been shown by Martínez and Zenk (2005) to be a frame. We show that the spectrum of this lattice is (homeomorphic to) precisely when is a -space. This we actually show to be true not only in spaces, but in locales as well. Recall that an ideal of a commutative ring is called a -ideal if whenever two elements have the same annihilator and...
Whitney C...-topologies and the Baire property.
Whitney maps-a non-metric case
It is shown that there is no Whitney map on the hyperspace for non-metrizable Hausdorff compact spaces X. Examples are presented of non-metrizable continua X which admit and ones which do not admit a Whitney map for C(X).
Zero-dimensional countable dense unions of Z-sets in the Hilbert cube
Zone and double zone diagrams in abstract spaces
A zone diagram of order n is a relatively new concept which was first defined and studied by T. Asano, J. Matoušek and T. Tokuyama. It can be interpreted as a state of equilibrium between n mutually hostile kingdoms. Formally, it is a fixed point of a certain mapping. These authors considered the Euclidean plane with finitely many singleton-sites and proved the existence and uniqueness of zone diagrams there. In the present paper we generalize this concept in various ways. We consider general sites...
Zur Topologisierung metrischer Räume. I.
Zwei Orbittheoreme für Bewegungsgruppen