On some properties of perfect images of GO-spaces and generalized trees

摘要

Let PIGO be the class of perfect images of generalized ordered (GO) spaces. We show that if X is an element of PIGO, then the following properties are equivalent: X is paracompact; X is metacompact; X is meta-Lindelof; X is subparacompact; X is submetacompact; X is weak ( delta theta) over bar -refinable; X is weak (theta) over bar -refinable; X is a D-space; X is an aD-space; X is transitively D; X is linearly D; No closed subspace of X is homeomorphic to a stationary subset of a regular uncountable cardinal; X is submeta-Lindelof. We also show that if f : X - Y is a perfect mapping such that Y is a semi-stratifiable space and X is a T1-space with a G(delta)*-diagonal, then X is a semi-stratifiable space. Let PIGT be the class of perfect images of generalized trees with Hausdorff generalized Sorgenfrey topologies. If X is an element of PIGT and X is a k-semistratifiable (stratifiable) space, then X is metrizable. If X is an element of PIGT, then X is first-countable if and only if X has a countable pseudocharacter. If f : L - X is a continuous irreducible mapping from a GO-space L (generalized tree L with a Hausdorff generalized Sorgenfrey topology) onto a space X with a regular G(delta)-diagonal (G(delta)-diagonal), then the space L has a regular G(delta)-diagonal (G(delta)*-diagonal). (C) 2020 Elsevier B.V. All rights reserved.