On topological spaces of orders (Q1030199)

Given a set \(X,\) the authors define the space \({ O}_p(X)\) of linear orders on \(X\) equipped with the topology of pointwise convergence as follows: The underlying set of \({ O}_p(X)\) is \({O}(X),\) the set of all linear orders on \(X.\) A basic neighbourhood at \(o\in {O}_p(X)\) is \(\{p\in {O}(X):p|_F=o|_F\}\) where \(F\subset X\) is finite. (Here for linear orders \(o,p\in { O}(X)\) and a set \(F\subset X\) they write \(o|_F=p|_F\) if the linear orders \(o\) and \(p\) coincide on \(F.\)) They show that if \(| X| =\omega_1\) or \(| X|=\omega_0\) then \({ O}_p(X)\) is homeomorphic to \(2^{\omega_1}\) and \(2^{\omega_0},\) respectively. One of the stated problems asks whether for any infinite cardinal \(\tau\), \(O_p(\tau)\) is homeomorphic to \(2^\tau.\)