Question on Basic Topology, open sets

  • Thread starter de1irious
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  • #1
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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!
 
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  • #2
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.
 
  • #3
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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.
 
  • #4
de1irious said:
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} ?
 
  • #5
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well yes, but also 23, 25, 41, etc
 
  • #6
HallsofIvy
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Let p be in E+ F. Then p= x+ y for some x in E, y in F. Since E is open, there exist [itex]\delta_1[/itex] such that the [itex]\delta_1[/itex] neighborhood of x is as subset of E. Since F is open, there exist [itex]\delta_2[/itex] such that the [itex]\delta_2[/itex] neighborhood of y is a subset of F. Now let [itex]\delta= \delta_1+ \delta_2[/itex] and show that the [itex]\delta[/itex] neighborhood of p= x+y is in E+ F.
 

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