Subspaces of the Sorgenfrey line and their products

Abstract

In this article we study the products of subspaces of the Sorgenfrey line 9. Using an idea by D. K. Burke and J. T. Moore[2] we prove in particular the following: Let $X_{i}, i = 1, \ldots ,n, n \geq 1$, be subspaces of $\mathscr{S}$, where each $X_i$ is uncountable. Then $X_{1} \times \ldots \times X_{n} \times x \mathscr{Q}$ can be embedded in $\mathscr{S}^{n+1}$ but can not be embedded in $\mathscr{S}^{n}$, where $\mathscr{Q}$ is the space of rational numbers with the natural topology. This statement strengthens [2, Theorem 2.1].