When is a Discrete Metric Space Compact?

In summary, the exercise asks to show that in a metric space with the discrete metric, X is always complete and the compactness of X depends on whether it contains a finite number of points or not. The definition of compactness in this exercise is that every sequence in X has at least one convergent subsequence. The Proposition states that in a compact metric space, X is both complete and bounded. To prove compactness, one can use the fact that in the discrete metric, every set is open and the collection of singleton sets can serve as an open cover for X. The exercise also suggests considering the existence of a finite subcover for X to be compact.
  • #1
mattmns
1,128
6
Here is the exercise:
----------
Let [itex](X,d_{disc})[/itex] be a metric space with the discrete metric.
(a) Show that X is always complete
(b) When is X compact, and when is X not compact? Prove your claim.
---------

Now (a) is pretty simple, but for (b) I am still not sure.

Here is our definition of compact: A metric space (X,d) is said to be compact iff every sequence in (X,d) has at least one convergent subsequence.

We also have the following Proposition: Let (X,d) be a compact metric space. Then (X,d) is both complete and bounded.

This tells us that we must have X bounded and complete (the latter we already have).

But I seem to be out of ideas. I almost feel as if the answer will be any X, or something of that nature (just from the way the exercise is written).

The entire exercise feels silly, as if I am just missing some silly detail that pulls the whole thing together. Any ideas? Thanks!

edit...

Since X needs to be bounded we know that there must be a ball B(x,r) in X which contains X. This would imply X is open (since X would be a ball), but that ... I am not sure. It just seems as though there is not much to go on here.

I was also maybe thinking about contradiction. Assuming that there is some sequence which has no convergent subsequence. But all of these feel strange without having any idea of what X is, or might be.
 
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  • #2
What's the discrete metric ? How is it defined ?
 
  • #3
What sort of sequences must X contain (or not contain) so that they have converging subsequences?
 
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  • #4
If you just write down what a convergent sequence is in the discrete metric you'll see the answer (and if not then look up one post ot gammamcc).
 
  • #5
Have you proved that, in the discrete metric, every set is an open set? In particular, every "singleton" set, every set containing exactly one ,is open. Given any set, X, the collection of all singleton sets, containing the points of X, is an open cover for X. Under what conditions does there exist a finite subcover?
 
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  • #6
That isn't the definition the OP has of compactness. I thought of posting it too, but they are using sequential compactness (which is of coiurse equivalent to the proper definition of compactness in a metric space).
 
  • #7
Thanks for the responses.

So X must contain a finite number of points to be compact.

The book has an exercise that says to prove that if every open cover of X has a finite subcover, then X is compact. Which I am currently messing around with.
 

What is a Discrete Metric Compact?

A Discrete Metric Compact is a type of metric space in which all points are isolated, meaning that each point has a neighborhood that contains only itself. This type of space is often used in topology and analysis.

How is Discrete Metric Compact different from other types of metric spaces?

In other metric spaces, points may be close to each other or even clustered together, but in a Discrete Metric Compact, all points are completely separated. This makes it a unique and useful tool for studying certain mathematical concepts.

What are some examples of Discrete Metric Compact spaces?

Some examples include the set of natural numbers with the discrete metric, the discrete topology on any finite set, and the Cantor set. These spaces have distinct points that are completely separated from each other.

What are the properties of Discrete Metric Compact spaces?

Discrete Metric Compact spaces have several important properties, including being compact, Hausdorff, and totally disconnected. These properties make them useful for studying limit points, continuous functions, and other mathematical concepts.

How is Discrete Metric Compact used in scientific research?

Discrete Metric Compact spaces have applications in various areas of science, such as physics, computer science, and biology. They can be used to model physical systems, analyze complex data sets, and study biological networks.

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