I have only just returned from holiday, and my brain may be bleached out by the sun. But my thinking is that the set is open for the product of the 1-dimensional finite complement topologies, but not for the 2-dimensional finite complement topology (because its complement obviously contains infinitely many points).
A projection map used in the above problem is open, but not continuous. So, I gave a subbasis element rather than or .
Although I was not able to prove that is a subbasis element for the above problem, I did not find any counterexample or contradiction that it is not a subbasis element for the above problem either.
Please correct me if I am wrong.
Now, going back to the original question,
"Is the finite complement topology on the same as the product topology on that results from taking the product of two finite complement topology? why?"
Since the projection maps onto the coordinate spaces are not continuous for the above problem, the product topology on by taking the product of two finite complement topology cannot even be defined in the first place.
Is that right?
In this example, the projection maps onto the coordinate spaces are not continuous when considered as maps from the space with the finite complement topology to the space with the finite complement topology. Therefore the finite complement topology on is different from the product topology on that arises from taking the finite complement topology on the two coordinate spaces. In fact, the product topology is stronger, since it contains more open sets.
If I am able to resolve the below contradiction I've found, I might be able to grasp the whole idea.
"If a product topology T on taking the product of two finite complement topology is defined, there exists a subbasis element for T that is not open".
According to the definition of a product topology on wiki, "...The product topology on X is defined to be the coarsest topology for which all the projections are continuous..".
If all the projections are continuous, a typical subbasis element defined by a product topology is open, and there is no contradiction.
To the best of my knowledge, there is no restriction that all the projections are continuous for a box topology.
Thus, we can define a box topology on taking the product of two finite complement topology. I think it is not the case for the product topology.
Please correct me if the above argument is wrong.
Thanks for all your comments so far. It is a great learning experience for me.