Unbounded towers and the Michael line topology

A topological space satisfies $\GNga$ (also known as Gerlits--Nagy's property $\gamma$) if every open cover of the space such that each finite subset of the space is contained in a member of the cover, contains a point-cofinite cover of the space. A topological space satisfies $\ctblga$ if in the above definition we consider countable covers. We prove that subspaces of the Michael line with a special combinatorial structure have the property $\ctblga$. Then we apply this result to products of sets of reals with the property $\GNga$. The main method used in the paper is coherent omission of intervals invented by Tsaban.