Metric spaces and closed balls

[Mathmos6]

Homework Statement

Can anyone suggest a simple example of a metric space which has a closed ball of radius, say, 1.001 which contains 100 disjoint closed balls of radius one?

I've taught myself about metric spaces recently so I'm only just getting started on it really, not really sure how to tackle this so any help would be very handy! Thanks!

 

[Mathmos6]

Would the discrete metric work, but defining it as 1.001 when x=/=y and 0 when x=y, on some set with over 100 members?

 

[Dick]

Sure, I think that works. You could even embed your example in euclidean space by picking your points at the vertices of a higher dimensional analog of a tetrahedron.

 

[Mathmos6]

Sure, I think that works. You could even embed your example in euclidean space by picking your points at the vertices of a higher dimensional analog of a tetrahedron.

Thanks very much! :)


Edit: Sorry, stuck again!

Q: Suppose that R × R is endowed with a product metric (pick your favourite). Show that the map f : R x R -> R defined by f(x,y) = x + y is continuous.

Attempt at solution: So I assume it means a product metric like d'((x1,x2),(y1,y2))=d(x1,y1)+d(x2,y2) and showing that f is continuous with respect to (d',d)? So then we want to show for all (a1,a2) and for all epsilon E there exists a delta D such that d(x1,a1)+d(x2,a2) <= D implies d(x1+x2,a1+a2) <= E, where d could just be any metric on the reals? Or do you think it means the Euclidean metric despite not saying so?

 

[Mathmos6]

(Also, should I have started a new thread for a new but related question? Apologies if so.)

 

[Dick]

I think you want to use the usual metric on R, d(x,y)=|x-y|. They appear to be letting you pick which norm to use the define the product metric. Opening a new thread for a new problem is a good idea because you will probably get more responses more quickly.

 

[Mathmos6]

I think you want to use the usual metric on R, d(x,y)=|x-y|. They appear to be letting you pick which norm to use the define the product metric. Opening a new thread for a new problem is a good idea because you will probably get more responses more quickly.

Okay, so (I'm assuming or hoping you know the result, or rather how general it is regarding which metrics to use!) do you think I use the metric |x-y| on R in f:R^2 -> R -and- in the 'd' of the product metric, or just the R to which f maps? Sorry I'm so unclear, the question is very vague and since I'm working about 12 lectures ahead of schedule, my lecturer hasn't mentioned or explained anything about it, although I doubt he will anyway! Thanks so much for the help!

 

[Dick]

The question seems to allow you to choose which product metric to use. Using d(x,y)=|x-y| on R, you could pick the product metric in various ways, p((x1,y1),(x2,y2))=(|x1-x2|+|y1-y2|) or sqrt(|x1-x2|^2+|y1-y2|^2) or max(|x1-x2|,|y1-y2|). etc. Just pick one.