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.