Proving that a subset is open

1. Sep 7, 2013

1. The problem statement, all variables and given/known data
Show that the subset D={(x,y)| x≠0 and y≠0} is an open set in R^2
.

2. Relevant 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.

3. 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).

Last edited: Sep 7, 2013
2. Sep 7, 2013

vela

Staff Emeritus
What does the notation f(x;y) mean?

3. Sep 7, 2013

Sorry about that, I miss quoted the question

4. Sep 7, 2013

vela

Staff Emeritus
First, tell us the definition of an open set.

5. Sep 7, 2013

CompuChip

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. Sep 7, 2013

Zondrina

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?

Last edited: Sep 7, 2013