Sequences of 0's and 1's: sequence spaces with the separable Hahn property
In  it was discovered that one of the main results in  (Theorem 5.2), applied to three spaces, contains a nontrivial gap in the argument, but neither the gap was closed nor a counterexample was provided. In  the authors verified that all three above mentioned applications of the theorem are true and stated a problem concerning the topological structure of one of these three spaces. In this paper we answer the problem and give a counterexample to the theorem in doubt. Also we establish a new way of constructing separable Hahn spaces.