Dual properties of monotonically normal spaces and generalized trees

Abstract

In the first part of this note, we show that every monotonically normal space is dually scattered of rank ≤2.In the second part of this note, we introduce a notion of a generalized tree with the Sorgenfrey topology (a generalized Sorgenfrey topology). A partially ordered set X is said to be a generalized tree if (←,x)={y∈X:y<x} is a linearly ordered set for each x∈X. We say that a generalized tree X has the Sorgenfrey topology (a generalized Sorgenfrey topology) if each x∈X with (←,x)=∅ is an isolated point, and for each x∈X with (←,x)≠∅, {(y,x]:y∈(←,x)} is a neighborhood base at x (or x is an isolated point).We get the following conclusions. A topological space X is monotonically normal and homeomorphic to some generalized tree with some generalized Sorgenfrey topology if and only if X is a topological sum such that each factor is homeomorphic to a linearly ordered set with a generalized Sorgenfrey topology.For a topological space X, the following are equivalent:(a)X is monotonically normal and homeomorphic to some generalized tree with the Sorgenfrey topology.(b)X is a topological sum such that each factor is homeomorphic to a linearly ordered set with the Sorgenfrey topology.(c)The condition (c1) or (c2) below holds.(c1)X is homeomorphic to some linearly ordered set with the Sorgenfrey topology.(c2)X is a topological sum having at least two, but finitely many factors, and each factor is homeomorphic to an ordinal of uncountable cofinality.We also get some conclusions on subspaces of ordinals which relate to a generalized tree with the Sorgenfrey topology.