# Metric spaces and closed balls

## 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!

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
Homework Helper
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.

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?

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

Dick