Monotone Lindelöf Property and Linearly Ordered Extensions

Abstract

In this paper, we explore the monotone Lindelöf property of two kinds of linearly ordered extensions of monotonically Lindelöf generalized ordered spaces. In addition, we construct nonseparable monotonically Lindelöf spaces using the Bernstein set, which generalizes Corollary 4 of Levy and Matveev ['Some more examples of monotonically Lindelöf and not monotonically Lindelöf spaces ', Topology Appl. 154 (2007), 2333-2343.2000 Mathematics subject classification: primary 54F05; secondary 54D20, 54D65.

Notes on monotone Lindelöf property

“…In Definition 2.2, if X, Y are LOTS, then τ X * Y = λ X * Y and if Y has two endpoints, X is a quotient space of GOTP(X * Y ) by Lemma 3.5 in [8]. For each x ∈ X, the subspace {x} * Y of GOTP(X * Y ) is homeomorphic to Y .…”

“…In particular, the double arrow space [0, 1] × {0, 1} with lexicographic order is monotonically Lindelöf. In [7], [8] we introduced a new topology on the lexicographic product set X × Y , where X, Y are generalized ordered (GO) spaces. This new topology contains the usual open-interval topology of the lexicographic order and also reflects in a natural way the fact that X and Y carry a GO-topology, rather than just the open interval topology of their linear orderings, which is called a generalized ordered topological product (GOTP) of the GO-spaces X and Y and is denoted by GOTP(X * Y ).…”

A Result on Monotonically Lindelöf Generalized Ordered Spaces

“…In this paper, we give a negative answer to this question. For undefined terms and notation we refer to [2,5,7].…”