Notes on the Collins Roscoe property and D-spaces

摘要

Let X be a space and W = {W(x) : x is an element of X}, where each W(x) is a countable family of subsets containing x. If X has W satisfying: if x is an element of U and U is open, then there exists an open V = V(x, U) containing x such that x is an element of W subset of U for some W is an element of W(y) whenever y is an element of V, we say that the space X satisfies the Collins-Roscoe property. For simplicity, we call the Collins-Roscoe property (G) in the paper. A neighborhood assignment for a space X is a function cc from X to the topology of X such that x is an element of (phi(x) for all x is an element of X. A space X is said to be a D-space if for every neighborhood assignment phi for X, there exists a closed discrete subset D of X such that X = U{phi(d) : d is an element of D}. Spaces satisfying (G) are D-spaces. We show the following. (1) The preimage of a (monotone) D-space under a fiber-discrete mapping is a (monotone) D-space. (2) Any uncountable subspace of the Sorgenfrey line and the Alexandroff double arrow space do not satisfy (G); the Niemytzki plane is not meta-Lindelof. (3) A space satisfies (G) if and only if its Alexandroff duplicate does so. (4) A locally separable regular space satisfying open (G) is metrizable; a locally quasi-developable, hereditarily meta-Lindelof space has a point-countable base. (5) The Tychonoff product of a D-space and a D-c-space is a D-space. (6) Let X and Y be linearly ordered spaces and Y have the maximal point and the minimal point and satisfy (G). Then the lexicographical ordered product X x Y with the ordered topology satisfies (G) if and only if the r-ordered space (X, T-r) and the l-ordered space (X, T-l) satisfy (G). (C) 2015 Elsevier B.V. All rights reserved.