Hi, we had an exam recently, and on it was this question:

Show that the order topology on {-1,0,1} x Z+ x Z is not the discrete topology.

It's a variation of the problem:

Show that the order topology on {-1,0,1} x Z+ is not the discrete topology.

That one I can show, and it's basically because Z+ has a minimum value of 1, so if you take an open set about 0 x 1 for example, then it will include some elements of the form -1 x n.

I'm having doubts about the exam question being valid, because it seems like it's also the discrete topology because of the Z part enabling us to get singletons. I could be wrong, really not sure ... but it's been disintegrating my brains for a while now