Are there any metric spaces with no Cauchy sequences?

In summary, a metric space is considered complete if all Cauchy sequences converge within it. However, it is possible for a metric space to have no Cauchy sequences, as in the case where points are sufficiently far apart from each other. In such cases, only eventually-constant sequences converge, and it can be said that sequences without repeated terms may not converge.
  • #1
dumb_curiosity
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0
A metric space is considered complete if all Cauchy sequences converge within the metric space. I was just curious if you could have a case of a metric space that doesn't have any Cauchy sequences in it. Wouldn't it be complete by default?

When trying to think of a space with no cauchy sequences, I invision a space where points are sufficiently far apart from each other. Since all metric spaces are T1,T2,T3, and T4, I get the idea that you can "separate" points, and closed sets in general in a metric space. That's what made me think about a metric space where points are so far apart, that there are no cauchy sequences.
 
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  • #2
You can always define Cauchy sequences by repeating a point : {p,p,...,p,...} . In discrete (and other) spaces , only eventually-constant sequences converge. So the best is to say that sequences without repeated terms may not converge.
 

Related to Are there any metric spaces with no Cauchy sequences?

1. What is a metric space?

A metric space is a set of points that are characterized by a distance function, also known as a metric. This function assigns a non-negative real number to every pair of points in the set, representing the distance between them.

2. What are Cauchy sequences?

A Cauchy sequence is a sequence of points in a metric space where the distance between any two points in the sequence becomes arbitrarily small as the sequence progresses. In other words, the points in the sequence get closer and closer together as the sequence goes on.

3. Can a metric space have no Cauchy sequences?

Yes, a metric space can have no Cauchy sequences. This means that there are no sequences of points in the space where the distance between any two points becomes arbitrarily small. In other words, there is no way to continuously zoom in on any point in the space.

4. What are some examples of metric spaces with no Cauchy sequences?

One example is the discrete metric space, where the distance between any two distinct points is always 1. Another example is the finite metric space, where the distance between any two points is less than or equal to the number of points in the space.

5. Why are metric spaces with no Cauchy sequences important?

Metric spaces with no Cauchy sequences are important because they help us understand the properties and limitations of metric spaces. They also have applications in areas such as topology and analysis. Additionally, studying these spaces can lead to the development of new mathematical concepts and techniques.

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