Subspaces of the Sorgenfrey line and their products (Q875684)

The main aim of this paper is to generalize a theorem of \textit{D. K. Burke} and \textit{J. T. Moore} [Topology Appl. 90, 57-68 (1998; Zbl 0967.54030)] which states that a product of \(n+1\) uncountable subspaces of the Sorgenfrey line \(S\) is not embeddable in \(S^{n}\). The main result of the paper (Theorem 1.1) states: Let \(\mathcal F\) be a finite family of non-empty subsets of \(S\) which are either uncountable or homeomorphic to the space of rationals \(\mathbb Q\) or discrete and let \(m\) be the number of uncountable elements of \(\mathcal F\), \(n\) be the number of elements of \(\mathcal F\) which are homeomorphic to \(\mathbb Q\) and \(p\) be the number of discrete elements of \(\mathcal F\), where \(m+n+p\geq 1\); then the product of all elements of \(\mathcal F\) can be embedded in \(S^{q}\) but not in \(S^{q-1}\), where \(q=m+1\) if both \(m\) and \(n\) are non-zero and \(q=m\) otherwise (\(S^{0}\) is defined to be \(\mathbb Q\) and \(S^{-1}\) to be empty). This result strengthens the aforementioned result of Burke and Moore since every uncountable subspace of \(S\) contains a homeomorphic copy of \(\mathbb Q\). A second theorem gives a characterization of which products of two and three subspaces of \(S\) can be embedded in \(S\) and \(S^{2}\), respectively.