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

Abstract

Let PIGO be the class of perfect images of generalized ordered (GO) spaces. We show that if X∈PIGO, then the following properties are equivalent: X is paracompact; X is metacompact; X is meta-Lindelöf; X is subparacompact; X is submetacompact; X is weak δθ‾-refinable; X is weak θ‾-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-Lindelöf. 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 Gδ⁎-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∈PIGT and X is a k-semistratifiable (stratifiable) space, then X is metrizable. If X∈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 Gδ-diagonal (Gδ⁎-diagonal), then the space L has a regular Gδ-diagonal (Gδ⁎-diagonal).