A note on products of (weakly) discretely generated spaces

In the last part of this note, we mainly get the following conclusions. If X is a locally compact GO-space and there exists a closed discrete set F such that for any point x∈X∖F, there is a neighborhood Vx of x in X such that Vx×Y is weakly discretely generated, where Y is a weakly discretely generated space, then X×Y is weakly discretely generated. If X is a locally compact GO-space such that X×Y is weakly discretely generated and lX∖X is scattered, where lX is a linearly ordered compactification of X and Y is a weakly discretely generated space, then lX×Y is weakly discretely generated.