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.

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In this paper, we show that the character of any monotonically Lindelöf generalized ordered (GO) space is not greater than ω 1 , which gives a negative answer to a question posed by Levy and Matveev ['Some questions on monotone Lindelöfness', Questions Answers Gen. Topology 26 (2008), 13-27, Question 51].

In this paper, we show that a generalized ordered space representable as the union of two closed monotonically Lindelöf subspaces is monotonically Lindelöf, which partially answers a question [7, Question 2] of Levy and Matveev. In addition, we show that the monotone Lindelöf property is hereditary with respect to open Lindelöf subsets in generalized ordered spaces. (2010): 54F05, 54D20. Mathematics Subject Classification

Abstract:In this paper, we show that any generalized ordered space X is monotonically (countably) metacompact if and only if the subspace X fxg is monotonically (countably) metacompact for every point x of X and monotone (countable) metacompact property is hereditary with respect to convex (open) subsets in generalized ordered spaces. In addition, we show the equivalence of two questions posed