We prove that if X is a space satisfying the discrete countable chain condition with a rank 3-diagonal then the cardinality of X is at most c.2010 Mathematics subject classification: primary 54D20; secondary 54E35.
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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