Open set x open set

1. Jun 18, 2011

seydunas

Hi

i want to solve a problem about topology or analysis: Let U and V be open sets in R^n, i want to to show their dot product is open in R.

Tahnk you

2. Jun 18, 2011

micromass

Hi seydunas!

The dot product is a composition of sums and products, so it actually suffices to show that sums and products are open. Can you show that?

3. Jun 18, 2011

seydunas

Hi Micromass,

i have showed that sums and products are open, i.e sum of two open sets and product of two open sets open, in fact sum of an open set and arbitrary set is open, ok, but i cant understand how it suffices to show their composition is also open. Can you explain it more detail.

4. Jun 18, 2011

micromass

Let $f:\mathbb{R}\times\mathbb{R}\rightarrow \mathbb{R}$ denote sum and let $g:\mathbb{R}\times\mathbb{R}\rightarrow \mathbb{R}$ denote product. Then, for example in $\mathbb{R}^2$

$$(x,y)\cdot (x^\prime,y^\prime)=xx^\prime+yy^\prime=f(g(x ,x^\prime),g(y,y^\prime))$$

So you see that this dot product is simply the composition of sum and products.

5. Jun 18, 2011

Bacle

Sorry if I am being thick here, but how do you define the dot product of open sets? Is it the union <a,B> for fixed a in A, and all b in B?

6. Jun 18, 2011

micromass

Yes, I took it as

$$\{<a,b>~\vert~a\in A,b\in B\}$$

7. Jun 19, 2011

HallsofIvy

I think you mean "Cartesian product", not "dot product".