# Thread: Is the product of two open sets open?

1. ## Is the product of two open sets open?

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?

2. Originally Posted by JohnStaphin
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?
really? it's true in $\mathbb{R} \times \mathbb{R}$? how so?

3. Originally Posted by Jhevon
really? it's true in $\mathbb{R} \times \mathbb{R}$? how so?
Let $(a,b) \in A\times B$. Now $a\in A$ and $b\in B$ so there is $(a-\epsilon,a+\epsilon)\subset A$ and $(b-\epsilon,b+\epsilon)\subset B$. Thus, $(a-\epsilon,a+\epsilon) \times (b-\epsilon,b+\epsilon) \subset A\times B$. Thus, the open square centered at $(a,b)$ is contained in $A\times B$. Therefore, $A\times B$ is open.

4. Originally Posted by ThePerfectHacker
Let $(a,b) \in A\times B$. Now $a\in A$ and $b\in B$ so there is $(a-\epsilon,a+\epsilon)\subset A$ and $(b-\epsilon,b+\epsilon)\subset B$. Thus, $(a-\epsilon,a+\epsilon) \times (b-\epsilon,b+\epsilon) \subset A\times B$. Thus, the open square centered at $(a,b)$ is contained in $A\times B$. Therefore, $A\times B$ is open.
ah, yes, thanks. i was thinking of open intervals as an example, but when i drew a picture to see if it were true, i treated the intervals as closed intervals

5. When you have two topological spaces and you want to create their product space, you need to define what is an open set.
usually the definition is that the topological basis is the set of products of open sets, meaning that in AxB the basis contains sets of the form UxV where U is open in A and V is open in B. any other open set in AxB is a union of sets from the basis.
so by this definition, a product of two open sets is an open set.