Is the product of two open sets open?

[Thorn]

Say we have open sets from two topological spaces. A is an open subset of T1 and B is an open subset of T2. So for these two open sets, A, B. Is A X B open itself? I see this is the case in R x R, where R is the real number line. I am wanting to say that this is just true in general... If so, anyone know a good proof? Or can anyway tell me how to show this?

 

[jostpuur]

The answer depends on what topology you give onto the set $$T_1\times T_2$$.

For example, if you let $$\mathbb{R}$$ have the standard topology, but give $$\mathbb{R}\times\mathbb{R}$$ a trivial topology $$\{\emptyset, \mathbb{R}\times\mathbb{R}\}$$, then $$]0,1[\times ]0,1[$$ is not open.

A standard choice for the topology of $$T_1\times T_2$$ is so called product topology. Here's the Wikipedia page of it: http://en.wikipedia.org/wiki/Product_topology
In product topology, $$A\times B$$ is open in $$T_1\times T_2$$ always when $$A$$ is open in $$T_1$$ and $$B$$ is open in $$T_2$$. This is almost by definition of the product topology. However, the precise definition is slightly complicated, so there is something left to be proved.