Prove set is open using open ball

[amanda_ou812]

Homework Statement

Prove that the set O = {(y1, y2) : y1 - y2 > 0} is an open subset of R2 in the Euclidean metric.

Homework Equations

The Attempt at a Solution

Try as I might I cannot figure out what to take for my r. Any suggestions?

 

[Dick]

Graph the set! It's bounded by the line y1=y2, right? Find the distance from an arbitrary point to that line.

 

[amanda_ou812]

I did graph it. It is the graph of y2 > y1. So on the coordinate plane it would be the dashed line y=x shaded above. I just guessed that my r might be (y1 +y2)/2 but when I try all my inequalities I can't get to x2>x1 (assuming that (x1, x2) is my other point in the ball that I need to show is in O)

 

[Dick]

Try http://mathworld.wolfram.com/Point-LineDistance2-Dimensional.html The actual distance of (y1,y2) to the boundary line is (y1-y2)/sqrt(2). (y1+y2)/2 can't possibly be right, it's not even positive.

 

[amanda_ou812]

Wait, but if (y1, y2) is in the open set, the y2-y1>0 which implies that y1-y2<0 so then wouldn't (y1-y2)/sqrt 2 be less than 0? I always have the hardest time with these open set questions

 

[Dick]

Wait, but if (y1, y2) is in the open set, the y2-y1>0 which implies that y1-y2<0 so then wouldn't (y1-y2)/sqrt 2 be less than 0? I always have the hardest time with these open set questions

Your original post said y1-y2>0. Seems to have gotten scrambled somewhere along the line. I would have a hard time proving x2<x1 just using basic inequalities. I would just draw the picture. Not good enough for your course??

 

[amanda_ou812]

Oh yes, the original post is incorrect. The set is y2-y1>0 or y2>y1. Sorry. No, a picture proof will not due. I am actually doing this as part of a larger problem. Let S be in R2 be the set defined by S = {(a1,a2) : 0 <= a2 <= a1}. Prove that S is a closed subset of R2 in the Euclidean metric. So, I was going about by proving that the complement is open. Well, the complement is the union of two open sets (which means the complement will be open). The first open set is x<0 which I was able to prove easily. The second set is y>x. Which I cannot figure out. Hence, the post. Grrr, the questions are so aggravating. once you think that you understand it, they throw out a question that stumps me.