Question on Basic Topology, open sets

[de1irious]

Hi,
In a euclidean space X with two subsets E and F, the subset E+F is defined as the collection of all x+y, where x E and y F. “+” denotes the addition in the euclidean space. Prove that if E and F are open, then E + F is open.

I'd really appreciate your help. Thanks so much!

 

[Gott_ist_tot]

Given that E and F are open for the remainder of this.

E + F = {x | x is an element of E or x is an element of F}

The definition the E + F given is also the definition of E union F.

Now it needs to be proved that the union of open sets is open. Your book should have a theorem and a proof of this so I will not prove it here due to lack of ability to write in proper notation.

 

[de1irious]

That's what I thought at first, but E+F isn't quite the union of E and F. In other words, if E is something like {1,3,5} and F is {20,40,60}, E+F does not contain any point in E or F. So I'm not sure that's the right proof.

 

[Gott_ist_tot]

That's what I thought at first, but E+F isn't quite the union of E and F. In other words, if E is something like {1,3,5} and F is {20,40,60}, E+F does not contain any point in E or F. So I'm not sure that's the right proof.

So, are you saying given those sets you defined E + F = {21, 43, 65} ?

 

[de1irious]

well yes, but also 23, 25, 41, etc

 

[HallsofIvy]

Let p be in E+ F. Then p= x+ y for some x in E, y in F. Since E is open, there exist $\delta_1$ such that the $\delta_1$ neighborhood of x is as subset of E. Since F is open, there exist $\delta_2$ such that the $\delta_2$ neighborhood of y is a subset of F. Now let $\delta= \delta_1+ \delta_2$ and show that the $\delta$ neighborhood of p= x+y is in E+ F.