Open Sets in R^n: Show Dot Product Is Open

[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

 

[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?

 

[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 can't understand how it suffices to show their composition is also open. Can you explain it more detail.

 

[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.

 

[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?

 

[micromass]

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?

Yes, I took it as

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

 

[HallsofIvy]

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