The existence of rigid measurable spaces (Q1118708)

For each uncountable cardinal \(\kappa\) the construction of \(2^{\kappa}\) linear orders \((S_ i,\leq)\) of cardinality \(\kappa\) is presented. Endowed with the order topology they have the following properties: they are pairwise nonembeddable into each other and the only embedding of \(S_ i\) into itself is the identity. If \(\kappa \geq \aleph_ 2\), then in each \(S_ i\) \(G_{\delta}\)-sets are open and hence the measurable spaces \((S_ i,{\mathcal B}_ i)\), where \({\mathcal B}_ i\) is the \(\sigma\)- algebra of all clopen subsets of \(S_ i\), are mono-rigid and pairwise nonembeddable into each other. Stationary sets control the character of points in \(S_ i's\). Some questions end the paper. (The reviewer has constructed a strongly rigid measurable space for \(\kappa =\aleph_ 1\), so the answer to one of them is known.)