Compare the dictionary order topology and product topology on RxR.

Question

I am not able to start this problem although I know the definitions for the basis of dictionary order topology on RxR and product topology on RxR.

Answer 1

HINT: Show first that if $U$ and $V$ are open sets in $\Bbb R$, then $U\times V$ is open in the dictionary order topology on $\Bbb R\times\Bbb R$; this is completely straightforward if you understand the definition of the dictionary order topology. Then use the fact that sets of the form $U\times V$ with $U$ and $V$ open in $\Bbb R$ are a base for the product topology to show that every set that is open in the product topology is also open in the dictionary order topology. Finally, find a set that is open in the dictionary order topology but not in the product topology; there are very simple examples.