# Hamming metric

1. ### Metric_Space

98
1. The problem statement, all variables and given/known data

I'm stuck on how to start this. The Hammin metric is define:
http://s1038.photobucket.com/albums/a467/kanye_brown/?action=view&current=hamming_metric.jpg

http://i1038.photobucket.com/albums/a467/kanye_brown/analysis_1.jpg?t=1306280360

a) prove the set U(d1,...,dp) is an open subset of X.

b) Prove that U is a basis of open sets for (X, d).

c) Say whether the statement is true or false.

Consider the following statement: “For every p 2 N and every d1, . . . , dp ϵ {0, 1},
the set U(d1,...,dp) is a closed subset of X.”
Is the statement true? Justify your answer (with a proof or counterexample).

Any ideas?

2. Relevant equations

3. The attempt at a solution

I'm not sure where to start. I know (X,d) is a metric space but that's about it so far.

2. ### micromass

18,695
Staff Emeritus
You can start by describing the balls in this space. Let's first answer an easy question:

given x=(0,0,0,...), can you find me all sequences y such d(x,y)<1/2?

3. ### Metric_Space

98
What I've figured out is the set is almost an open ball around the sequence d1,d2,d3...d_p of

I think there may be a special case where everything is the same except for x_p

4. ### micromass

18,695
Staff Emeritus
Well, maybe your open set is the union of two balls?

5. ### Metric_Space

98
I'm not sure I fully follow -- could you elaborate?

6. ### micromass

18,695
Staff Emeritus
Well, can you first tell me what the ball around (0,0,0,...) with radius 1/2 looks like?

7. ### Metric_Space

98
not sure...just a point?

8. ### micromass

18,695
Staff Emeritus
No, you'll need to find all $$(x_1,x_2,x_3,...)$$ such that

$$\sum_{i=1}^{+\infty}{\frac{x_i}{2^i}}<\frac{1}{2}$$

For example, (0,1,0,0,...) is such a point, (0,1,0,1,0,1,0,1,...) too...

9. ### Metric_Space

98
Still a bit stuck...

10. ### micromass

18,695
Staff Emeritus
OK, let's investigate the situation. For which vectors (x1,x2,x3,...) does

$$\sum_{k=0}^{+\infty}{\frac{x_k}{2^k}}=1$$?

11. ### Metric_Space

98
I guess ones whose infinite sum is one, ie. Things of the form 1/2 + 1/4 + ....

Not sure how to get that back into the corresponding x_k

12. ### micromass

18,695
Staff Emeritus
Well, once you figured that out, we can take the next step!

13. ### Metric_Space

98
Thanks -- what about b) ...

I don't really understand the concept of basis of open sets

14. ### micromass

18,695
Staff Emeritus
To solve (b), you will also need to know what the open balls look like. So you'll need to figure that out first.

A basis is just a collection of open sets, such that every open set can be shrunk to a basis open set. More formally: for every x and for every set G around x, we can find a basis element B such that $x\in B\subeteq G$.

The trick is that we don't need to show this for open sets in general (this would be too hard), but it suffices to look at balls. But of course, you'll need to know the balls for that...

15. ### Metric_Space

98
...care to give me a hint on what the balls look like?

I must admit, I find the notation confusing

16. ### micromass

18,695
Staff Emeritus
Well, I want to figure out together what the balls look like. But for that, you'll first need to answer my post 10, since this is the first to find what the balls look like...

17. ### Metric_Space

98
Is it points with the only the even entries allowed to equal to 1 and the odd entries equal to 0?

so the 2nd, 4th, 6th, etc could be 1 or 0
but 1st, 3rd, 5th, etc can only be 0?

18. ### micromass

18,695
Staff Emeritus
No, because (0,1,0,1,0,1,0,1,...) would correspond to

$$\sum_{k=1}^{+\infty}{\frac{1}{2^{2k}}=\sum_{k=1}^{+\infty}{\frac{1}{4^k}}=\frac{1}{3}$$

and this is not 1.

19. ### Metric_Space

98
Hmm...I'm stuck

20. ### Metric_Space

98
any suggestions on materials I can read? I'm not sure I can visualize this