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Proving that a subset is open

  • #1

Homework Statement


Show that the subset D={(x,y)| x≠0 and y≠0} is an open set in R^2
.

Homework Equations


Open set: U is a subset of R^n. U is an open set when for every point X1, contained within U, there exists some open disk centered around X1 with radius r>0, that is completely contained within U. Or for simplicity's sake, a set U is open if it does not contain any of its boundary points.


The Attempt at a Solution


I have an understanding of what makes up an open set and know why this set is open, but i have no idea as to how I am meant to prove this, graphically or analytically( ideal method).
 
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Answers and Replies

  • #2
vela
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Homework Statement


Show that the subset D={f(x; y)| x≠0 and y≠0} is an open set in R^2
.

Homework Equations





The Attempt at a Solution


I have an understanding of what makes up an open set and know why this set is open, but i have no idea as to how I am meant to prove this, graphically or analytically( ideal method).
What does the notation f(x;y) mean?
 
  • #3
Sorry about that, I miss quoted the question
 
  • #4
vela
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First, tell us the definition of an open set.
 
  • #5
CompuChip
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So let (x, y) be a point in D. Can you find a radius r such that all points in the open ball of radius r around this point are in D?

First you may want to try drawing this, then proving it analytically. Note that for most points it's pretty trivial - the interesting case is for when (x, y) is close to (0, 0).
 
  • #6
Zondrina
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So let (x, y) be a point in D. Can you find a radius r such that all points in the open ball of radius r around this point are in D?

First you may want to try drawing this, then proving it analytically. Note that for most points it's pretty trivial - the interesting case is for when (x, y) is close to (0, 0).
Also, presuming you can always find ##r## for arbitrary ##(x,y) ≠ (0,0)##, what happens at ##(0,0)## exactly?

Get reeeeeally close to ##0##. Can you always find an ##r## such that ##(0,0)## is not contained any neighborhood of your point?
 
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