Metric spaces and closed balls

In summary, the conversation discusses the concept of metric spaces and a request for a simple example of a metric space with specific properties. The conversation also touches on the continuity of a map in relation to a product metric. The expert suggests using the usual metric on R, d(x,y)=|x-y|, for the problem and clarifies that the question allows for different choices in the product metric.
  • #1
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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!
 
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  • #2
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?
 
  • #3
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.
 
  • #4
Dick said:
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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  • #5
(Also, should I have started a new thread for a new but related question? Apologies if so.)
 
  • #6
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.
 
  • #7
Dick said:
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!
 
  • #8
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.
 

What is a metric space?

A metric space is a mathematical concept that describes a set of objects along with a distance function that measures the distance between any two objects in the set. This distance function satisfies certain properties, such as being non-negative and symmetric.

What is a closed ball in a metric space?

A closed ball in a metric space is a subset of the metric space that includes all points within a given distance from a fixed point, called the center of the ball. The distance used is determined by the distance function of the metric space. A closed ball is defined by the condition that the distance between any point in the ball and the center is equal to or less than the given distance.

What is the difference between an open ball and a closed ball?

An open ball in a metric space is a subset that includes all points within a given distance from a fixed point, excluding the points on the boundary of the ball. In contrast, a closed ball includes all points on the boundary as well. Another way to think about it is that an open ball is like an empty sphere, while a closed ball is like a filled sphere.

What are some examples of metric spaces?

Some examples of metric spaces include Euclidean space, where the distance function is the usual distance formula, and the space of real-valued continuous functions on a closed interval, where the distance function is defined by the supremum norm. Other examples include discrete metric spaces, where the distance between any two points is either 0 or 1, and the space of square matrices with a given size, where the distance function is defined by the Frobenius norm.

How are closed balls used in mathematics?

Closed balls are used in many areas of mathematics, including analysis, topology, and geometry. They are useful for proving theorems and solving problems related to convergence, continuity, compactness, and completeness. They are also used in applications such as optimization and computer graphics. In addition, closed balls are fundamental objects in the study of metric spaces and provide a way to define neighborhoods and open sets, which are important concepts in topology.

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