On the lexicographic ordered spaces

In this paper, we discuss the lexicographic ordered topologies on some products. We mainly prove: (1) The lexicographic ordered space λγ is base-normal for any ordinals λ and γ. (2) The lexicographic ordered space [0,1)γ is Lindelöf for each ordinal γ≤ω1. (3) The lexicographic ordered space λγ is Lindelöf for any λ<ω1 and γ≤ω1.