1. ## Set Products Question

I'm beginning a section on Product Topologies and I have to do a few problems, but I'm not seeing the difference between these two.

Let $\mathcal{A}$ and $\mathcal{B}$ be two nonempty families of sets. Prove that:

$\bigcap \mathcal{A} \times \bigcap \mathcal{B} = \bigcap\{A \times B : A \in \mathcal{A}, \ B \in \mathcal{B}\}$

Suppose that $A_1,\cdots ,A_n$ and $B_1,\cdots ,B_n$ are sets. Prove that:

$(A_1\times B_1)\cap\cdots\cap (A_n\times B_n) = (A_1\cap\cdots\cap A_n) \times (B_1\cap\cdots\cap B_n)$

Aren't these asking for the same thing??

2. ## Re: Set Products Question

Hey Aryth.

Does your combination contain all permutations Ai X Bj for all possible i and j or is i = j? (The way you have written it is unclear).

3. ## Re: Set Products Question

To be honest I can't answer that question. The problems are exactly as I have typed them. In the proofs... I imagine that the equations would hold either way (Sets under intersections commute, so I can order them however I like on the right hand side and arrive at any necessary permutation as long as it's $A_i \times B_j$).

4. ## Re: Set Products Question

Originally Posted by Aryth
I'm beginning a section on Product Topologies and I have to do a few problems, but I'm not seeing the difference between these two.

Let $\mathcal{A}$ and $\mathcal{B}$ be two nonempty families of sets. Prove that:

$\bigcap \mathcal{A} \times \bigcap \mathcal{B} = \bigcap\{A \times B : A \in \mathcal{A}, \ B \in \mathcal{B}\}$

Suppose that $A_1,\cdots ,A_n$ and $B_1,\cdots ,B_n$ are sets. Prove that:

$(A_1\times B_1)\cap\cdots\cap (A_n\times B_n) = (A_1\cap\cdots\cap A_n) \times (B_1\cap\cdots\cap B_n)$

Aren't these asking for the same thing??
No, they are not. The second assumes a finite number of sets (n) while the first does not.