Proving A is Open in (R2,d): A Math Challenge

[e(ho0n3]

Homework Statement
Prove that the set A = {(x,y) in R² : xy ≠ 1} is open in the metric space (R², d), where d is the Euclidean metric.

The attempt at a solution
A is open if for any p in A, I can find an open ball centered at p that is contained in A. This would be easy if I could find the closest point q = (x,y) to p that is not in A. Then I would just choose the open ball whose radius is d(p,q). Heck, I don't even need to find q: It suffices to know that such a q exists. This is where I'm stumped. Any tips?

 

[morphism]

How about showing that the complement of A is closed? If you use sequences this should be pretty straightforward.

 

[e(ho0n3]

True, but this problem is given before the problems on sequences, so I think that would be unfair.

 

[morphism]

Fine. Let's take an element p=(a,b) in A, and suppose that every neighborhood of p contains elements of the form (x,1/x) for some (nonzero) x in R. This means that we can make the quantity |(a,b) - (x,1/x)|^2 arbitrarily small by choosing an appropriate x. But |(a,b) - (x,1/x)|^2 = (a-x)^2 + (b-1/x)^2, and both (a-x)^2 and (b-1/x)^2 are nonnegative, so ...

 

[e(ho0n3]

It seems to me that I must find the minimum of (a-x)² + (b-1/x)², but that requires calculus, which I'm not allowed to use yet. I can't think of anything else I can do here.

 

[Dick]

I don't know where you are getting these restrictions on what you can and cannot use. You HAVE to use something. There are continuous curves in R^2 that are dense. What can you use? Why can't you use calculus?

 

[Hurkyl]

It seems to me that I must find the minimum of (a-x)² + (b-1/x)², but that requires calculus, which I'm not allowed to use yet. I can't think of anything else I can do here.

Even if you can't use calculus in your proof, you can use it in your scratchwork. Once you know what the answer is supposed to be, it shouldn't be difficult to prove it with ordinary algebra of inequations.

 

[e(ho0n3]

Let's see, I can use the fact that the reals are a complete ordered field, the definition of metric space, the Schwarz inequality, the inequality

$$\sqrt{(a_1+b_1)^2 + \cdots + (a_n+b_n)^2} \le \sqrt{a_1^2 + \cdots + a_n^2} \, + \, \sqrt{b_1^2 + \cdots + b_n^2}$$,

the inequalities

$$d(p_1,p_n) \le d(p_1,p_2) + \cdots + d(p_{n-1},p_n)$$ and

$$|d(p,r)-d(r,q)| \le d(p,q)$$

the definitions of open/closed set/ball and the fact that metric spaces induce a topology.

 

[Dick]

Let's see, I can use the fact that the reals are a complete ordered field, the definition of metric space, the Schwarz inequality, the inequality

$$\sqrt{(a_1+b_1)^2 + \cdots + (a_n+b_n)^2} \le \sqrt{a_1^2 + \cdots + a_n^2} \, + \, \sqrt{b_1^2 + \cdots + b_n^2}$$,

the inequalities

$$d(p_1,p_n) \le d(p_1,p_2) + \cdots + d(p_{n-1},p_n)$$ and

$$|d(p,r)-d(r,q)| \le d(p,q)$$

the definitions of open/closed set/ball and the fact that metric spaces induce a topology.

How does that prove it?

 

[e(ho0n3]

How does that prove it?

Those are just some facts that I can use.

 

[Dick]

Ah, gotcha. Now why can't you use calculus or sequential closure, exactly? I think you are limiting yourself unfairly to the point where you can't prove it at all. Those properties are too vague to cover 1=xy.

 

[e(ho0n3]

I'm assuming this problem is from the section called Open and Closed Sets since some of the problems before and after it involve proving that some set is open or closed. The next section is called Convergent Sequences, so I'm assuming the author is expecting me to use only those notions prior to this section to solve the stated problem. This doesn't include calculus since the aim of the book is to develop calculus rigorously--and there is always the danger of circular arguments.

Anywho, I don't even understand how the sequential argument would work: Let B be the complement of A. I can show that B is closed by showing that any sequence of points in B converges to a point in B. But how is this true for the sequence (1,1), (2,1/2), (3,1/3)..., which goes to $(\infty,0) \not\in B$?

 

[e(ho0n3]

Also, the derivative of (a-x)² + (b-1/x)² is -2a + 2x + 2b/x² - 2/x³. I don't know how to find the zeros of that.

 

[Dick]

I'm assuming this problem is from the section called Open and Closed Sets since some of the problems before and after it involve proving that some set is open or closed. The next section is called Convergent Sequences, so I'm assuming the author is expecting me to use only those notions prior to this section to solve the stated problem. This doesn't include calculus since the aim of the book is to develop calculus rigorously--and there is always the danger of circular arguments.

Anywho, I don't even understand how the sequential argument would work: Let B be the complement of A. I can show that B is closed by showing that any sequence of points in B converges to a point in B. But how is this true for the sequence (1,1), (2,1/2), (3,1/3)..., which goes to $(\infty,0) \not\in B$?

You only have to show that the limits of CONVERGENT sequences are in B. I.e. if (a_i,1/a_i) converges to (b,c) can you show b=1/c? It's the same as the proof f(x)=1/x is continuous except at 0.

 

[e(ho0n3]

Oops. That's right. Suppose (a_i, 1/a_i) converges to (b, c). Then a_i converges to b and so 1/a_i converges to 1/b, hence c=1/b.

That was easy. What about finding the minima?

 

[Dick]

I guess I can't think of a good way to compute or estimate the minimum distance. But now we don't have to, right?

 

[e(ho0n3]

No. We don't have to. It would be interesting though.