Productivity of paracompactness in the class of the first-countable GO-spaces with certain dense subsets

Let P be the class of all spaces whose Cartesian product with every paracompact space is paracompact. We prove that if X is a paracompact, first-countable GO-space with σDC dense subset then X ∈ P if and only if X is σDC. We also prove that if Y is a metric GO-space and X is a GO-space defined on Y then X is metrizable provided X ∈ P . These results support the validity of the Telgárskyʹs conjecture which says that if X is a paracompact space then X ∈ P if and only if the first player of the G ( DC , X ) game introduced by R. Telgársky, see [13,4,5] has a winning strategy.