on $\displaystyle R^2$ we define an order topology (lexicographic order) as follow: $\displaystyle (x1,y1)>(x2,y2)$ if $\displaystyle x1>x2$ or ($\displaystyle x1=x2$ & $\displaystyle y1>y2$).

You have defined no topology at all: you've defined a (partial) order on $\displaystyle \mathbb{R}^2$ .

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.

Tonio
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)?

thanks