I think I understand this example from the book. I just need to verify.
Let . The dictionary order on is just the restriction to of the dictionary order on the plance . However, the dictionary order topology is not the same as the subspace topology on obtained from the dictionary order on . For example, the set is open in in the subspace, but not in the order topology.
Ok so the subspace is of the form where
Now, the intersection is , but why is this set considered open in the subspace topology when it is have closed?
However, the order topology only has half closed or open sets if the member is the least or largest element respectively. The largest element would be . Therefore, the set isn't open in the order topology.