A note on products of (weakly) discretely generated spaces

In the first part of this note, we mainly get the following conclusions. If X is a regular space with a nested local base at a point a of X and Y is discretely generated at a point b of Y, then X × Y is discretely generated at 〈 a , b 〉 . We finally show that if X is a GO-space and Y is a discretely generated space then X × Y is discretely generated. 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 V x of x in X such that V x × 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 l X ∖ X is scattered, where lX is a linearly ordered compactification of X and Y is a weakly discretely generated space, then l X × Y is weakly discretely generated.