# Homework Help: Metric Space/Open Ball

1. Sep 3, 2011

### muzak

Hi guys, two problems, first one I understand for the most part, the second one, I do not know how to set up and solve.

1. The problem statement, all variables and given/known data
Let X = R$^{n}$ for x = (a$_{1}$,...,a$_{n}$) and y = (b$_{1}$,...,b$_{n}$), define
d$_{\infty}$(x,y) = max {|a$_{1}$-b$_{1}$|,...,|a$_{n}$-b$_{n}$|}. Prove that this is a metric.

2. Relevant equations
Just the triangle inequality part for this one.

3. The attempt at a solution
I've proven the first 3 properties, not quite sure on the last part.
My attempt was to break it up into n cases by supposing a single
difference as the max in each case but with ellipsis but I wasn't
sure on the exactly how. Here's what I attempted:

The distance between the two points is the largest of the n cases.
Let z = (z$_{1}$,...,z$_{n}$). Then we have n cases to check.
Case 1: d(x,y) = |a$_{1}$-b$_{1}$|
Notice that |a$_{1}$-z$_{1}|$|$\leq$max{|a$_{1}$-z$_{1}$|,
(I stopped here because I wasn't sure how to proceed,
should I write out to n cases with the ellipsis or two cases?)
What I was going to show after this was that the triangle inequality
holds for a$_{1}$, a$_{2}$, ..., a$_{n}$.

1. The problem statement, all variables and given/known data
Prove that the set S = {(x$_{1}$,y$_{1}$ : x$_{1}$ + y$_{1}$ > 0}
is an open subset of R$^{2}$ in the Euclidean metric.

2. Relevant equations
Euclidean metric, Schwarz Inequality?, Open Ball.

3. The attempt at a solution
I'm not sure how to proceed with this one at all. Picture-wise,
it'd be the region above the y=-x line, I'm guessing,
and I'm guessing I have to pick some arbitrary point in that
region and calculate a strict inequality to show that this is open.
But I do not know how to proceed at all, I'm looking for some hand-holding
at this point really because I want to understand it step by step.

Thanks for any help!

2. Sep 3, 2011

### micromass

Let's prove this one like this. Can you first show that

$$|a_k-b_k|\leq \max \{|a_1-z_1|,...,|a_n-z_n|\}+\max \{|z_1-b_1|,...,|z_n-b_n|\}$$

??

OK, what does open mean for you?? You need to show for al (x,y) in the set that there exists a ball around(x,y) that stays in the set. How would you choose that ball??
Does it makes sense what I'm saying?

3. Sep 3, 2011

### muzak

$|a_k-z_k+z_k-b_k|$$\leq |a_k-z_k|+|z_k-b_k|\leq$ max${|a_1-z_1|,...,|a_n-z_n|}$+max{|$z_1-b_1|,...,|z_n-b_n$|}?

I guess for me right now, open for is something without its boundary points. I would have to pick an r such that for some $(x_2, y_2), (x_1, y_1) then (x_2-x_1)^{2} + (y_2-y_1)^{2} \leq r^{2}$?

Last edited: Sep 3, 2011
4. Sep 3, 2011

### micromass

OK, and now that the maximum of the left-hand side.

What is your definition of open? Saying that it's something without boundary isn't good enough unless you specify what a boundary point is.

5. Sep 3, 2011

### muzak

The max's are the distances, gotcha, thanks.

In this problem, I'd say the boundary is the y = -x line.

6. Sep 3, 2011

### micromass

Yes, but what is the definition of a boundary??

7. Sep 3, 2011

### muzak

Set of points that encloses a space, endpoints? I don't know what you mean.

8. Sep 3, 2011

### micromass

Please, look up in your course the definition of "open set" and such things. We can't really do much unless we have some definitions to work with...

9. Sep 3, 2011

### muzak

It's just the open ball centered at x

B(x,r) = { y in X: d(x,y) strictly less than r}

10. Sep 3, 2011

### micromass

What is just the open ball??

Could you please quote the exact definition of an open set?? Yes, it has something to do with open balls.

11. Sep 3, 2011

### muzak

Just quoting the definition from my notes:

Given a metric space (X,d) a subset S$\subseteq$X is called open provided that whenever x$\epsilon$S, then there is an r>0 such that B(x;r)$\subseteq$S.

12. Sep 3, 2011

### micromass

Exactly!!

So for our $S=\{(x,y)~\vert~x+y>0\}$, we need to find for each $(x,y)\in S$, a r such that $B(x,r)\subseteq S$.

So, let's pick an arbitrary (x,y) in S. What do you think we should take as our r?

13. Sep 3, 2011

### micromass

By the way, did you see the open set characterization of continuity yet?? That might come in very handy!!

14. Sep 3, 2011

### muzak

Something greater than 0 but less than x+y? I don't know how we go about picking the r.

I'm just starting out in real analysis, haven't come across it yet, we've only gotten past closed sets which was a bit easier to digest.

15. Sep 3, 2011

### micromass

OK, take an arbitrary point in S. What is the distance between that point and the line y=-x??

16. Sep 3, 2011

### muzak

$\sqrt{(y_{o}-y)^{2}+(x_{0}-x})^2$, since y=-x, do I make that substitution into the formula?

17. Sep 3, 2011

### micromass

What is $(x_0,y_0)$ and (x,y)??

18. Sep 3, 2011

### muzak

(x,y),(xo,yo) in S

19. Sep 3, 2011

### micromass

But why would you do that?? Why would you pick both of them in S??

I want to find the least distance between a point (x,y) in S and the line y=-x. This is just a calculus/geometry question... How would you solve that??

20. Sep 3, 2011

### muzak

I was thinking we had to use the eclidean distance formula for some reason. |x+y|/sqrt(2)

21. Sep 3, 2011

### micromass

Indeed, the distance between a point $(x,y)\in S$ and the line y=-x is exactly

$$\frac{|x+y|}{\sqrt{2}}$$

Now, if we choose

$$r=\frac{|x+y|}{\sqrt{2}}$$

isn't it true that the ball B((x,y),r) stays inside of S?

22. Sep 4, 2011

### muzak

Sorry for the departure.

Yes, because 0<$\frac{|x+y|}{\sqrt{2}}$<y1+y2. Thus B((x,y),r)$\subseteq$S proving the set is an open subset in R$^{2}$.

I see. So when the question asks take the Euclidean metric, I shouldn't use the euclidean metric we learned in class but just find the distance from previous knowledge?

So if I had to prove the set S={(y1,y2): y1>0} is an open subset of R$^{2}$.
Then all I'd have to do is let (x,y)$\epsilon$S and pick r = |x|, then B((x,y),r)$\subseteq$S making it an open subset?

23. Sep 4, 2011

### micromass

You did use the Euclidean metric here. To find out the distance between the line and the poitn, you actually needed to use the Euclidean metric/

The radius depends on the open set in question. Right now $|x+y|/\sqrt{2}$ works, but such a trick doesn't always work...

24. Sep 4, 2011

### muzak

Did I mess up the second one then?

I have a question on notation. My professor uses d$_{\infty}$(x,y) = max{|stuff|, ...}. So when the question says prove that the disk x$^{2}$+y$^{2}$<1 is an open subset of (R$^{2}$,d$_{\infty}$), what does (R$^{2}$,d$_{\infty}$) mean?