On the covering dimension of subspaces of product of Sorgenfrey lines

Abstract

Let S denote the Sorgenfrey line. Then the following results are proved in this paper: (i) If X is a nonempty subspace of S ℵ 0 {S^{{\aleph _0}}} , then dim ⁡ X = 0 \dim X = 0 . (ii) For any nonempty separable space X ⊂ S ℵ 0 , dim ⁡ X m = 0 X \subset {S^{{\aleph _0}}},\dim {X^m} = 0 for any cardinal m.