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Open set

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Homework Statement



Show that the set of [tex]\mathbb{R}^2[/tex] given by [tex]S = \{(x, y) \in \mathbb{R}^2 : x > y\}[/tex] is open.

Homework Equations





The Attempt at a Solution



Why is S the half plane that has boundary given by the line [tex]y = -x?[/tex]
 

Answers and Replies

  • #2
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Homework Statement



Show that the set of [tex]\mathbb{R}^2[/tex] given by [tex]S = \{(x, y) \in \mathbb{R}^2 : x > y\}[/tex] is open.

Homework Equations





The Attempt at a Solution



Why is S the half plane that has boundary given by the line [tex]y = -x?[/tex]
It's not - the boundary is the line y = x. I suspect a typo.
 
  • #3
Zondrina
Homework Helper
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It's not - the boundary is the line y = x. I suspect a typo.
This is what I also thought when I read the problem.

S is open if and only if S=S°.
 
Last edited:
  • #4
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Thanks for the clarification. I am still having trouble with this problem.

I know I must determine a radius r such that [tex]\{(x_1, y_2)\in \mathbb{R}^2: \sqrt{|x-y|^2 +|x_1-y_1|^2}<r\}[/tex]

but how can I find such an r?

What I did was:

I located the region of the plane where [tex]x\gt y[/tex] then I place a point [tex]P=(x_1,y_1)[/tex] in it where it is fairly close to the line [tex]y=x[/tex].

Now I moved to the left from the point P until I hit the line y=x, which I marked Q then this Q has coordinates (y_1,y_1).

But I cant find an r. The answer given is [tex]\frac{x_1 - y_1}{\sqrt{2}}.[/tex]

Which I don't know how they got?
 
  • #5
33,273
4,982
Thanks for the clarification. I am still having trouble with this problem.

I know I must determine a radius r such that [tex]\{(x_1, y_2)\in \mathbb{R}^2: \sqrt{|x-y|^2 +|x_1-y_1|^2}<r\}[/tex]
That equation doesn't look right to me. Why is it (x1, y2), and why doesn't y2 show up on the right?
but how can I find such an r?

What I did was:

I located the region of the plane where [tex]x\gt y[/tex] then I place a point [tex]P=(x_1,y_1)[/tex] in it where it is fairly close to the line [tex]y=x[/tex].

Now I moved to the left from the point P until I hit the line y=x, which I marked Q then this Q has coordinates (y_1,y_1).

But I cant find an r. The answer given is [tex]\frac{x_1 - y_1}{\sqrt{2}}.[/tex]

Which I don't know how they got?
 

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