.on we define an order topology (lexicographic order) as follow: if or ( & ).
You have defined no topology at all: you've defined a (partial) order on .
Now, as far as I know, an order topology is defined on a totally ordered set, so if you meant
this then you first prove the above gives you a total order on the real plane and then look at
the derived (order) topology, and then you can ask yourself whether this toplogy is the
same as the usual Euclidean one.
Is this topology equivalent to the standard topology? if not, which topology is finer?(give an example of an open set in one but not in the other)?