Can you think of a counter example (real analysis)

[robertdeniro]

Homework Statement

X and Y are two closed non-empty subsets of R (real numbers).

define X+Y to be (x+y | x belongs to X and y belongs to Y)

give an example where X+Y is not closed

Homework Equations

The Attempt at a Solution

i tried X=all integers and Y=[0 1] but that didnt work out.
i know X+Y must be non-finite because all finite subsets are closed.

 

[╔(σ_σ)╝]

Well you simply want to define X+Y such that some points are deleted.:-)

 

[robertdeniro]

Well you simply want to define X+Y such that some points are deleted.:-)

ya i was thinking of that too. haven't found a way yet? can you give 1 more hint?

 

[╔(σ_σ)╝]

Well define one point in X to be negative the other point in Y.

 

[robertdeniro]

Well define one point in X to be negative the other point in Y.

hmm just to make sure you didnt misunderstand the definition of X+Y

if X=[1 3] and Y={-2}

then X+Y = [-1 1]

so I am still confused by your suggestion

 

[╔(σ_σ)╝]

My suggestion is a bit vague and not helpful but think about if it is possible to delete stuff from X+Y.

 

[ystael]

Think about isolated points. Sets all of whose points are isolated are closed, but by manipulating them, you may be able to produce a limit point in a convenient place.

 

[robertdeniro]

thanks for all the tips, I am going to think about this, hopefully ill discover it soon

 

[robertdeniro]

i still haven't figure this out yet...but i was wondering if the solution has a "hole" in an interval?

and I am still confused about the "deleting" suggestion. since a negative and a positive would only produce zero and not remove the element from the set. in essence, you can only stretch and shift an interval.

 

[Dick]

i still haven't figure this out yet...but i was wondering if the solution has a "hole" in an interval?

and I am still confused about the "deleting" suggestion. since a negative and a positive would only produce zero and not remove the element from the set. in essence, you can only stretch and shift an interval.

Take r to be irrational. Did you know you can find whole multiples of r that are arbitrarily close to being integers?

 

[robertdeniro]

i think if X=the integers and Y=sqrt(2) * the integers then it would work.

i still can't think of an example where points are deleted.

 

[Dick]

i think if X=the integers and Y=sqrt(2) * the integers then it would work.

i still can't think of an example where points are deleted.

If you think the first example will work can you show it does? I wouldn't worry about the 'points deleted' example. It's not at all clear to me what that's supposed to mean.