On discretely generated box products

A topological space X is called discretely generated   if for any A⊆XA⊆X and x∈A‾ there exists a discrete set D⊆AD⊆A such that x∈D‾. We solve the Problems 3.19 and 3.3 in [2]. Problem 3.19: Does the space {ξ}∪ω{ξ}∪ω embed into a box product of real lines when ξ∈βω∖ωξ∈βω∖ω? For any ξ∈βω∖ωξ∈βω∖ω, we answer negatively. Problem 3.3: Is any box product of first countable spaces discretely generated? We answer positively by assuming that the spaces are regular.