# Complete Metric Spaces .... Conway, Analysis, Section 5.2 ....

• MHB
• Math Amateur
In summary, Peter is reading John B. Conway's book "A First Course in Analysis" and is focused on Chapter 5: Metric and Euclidean Spaces, particularly Section 5.2: Sequences and Completeness. Peter needs clarification on Conway's definition of completeness of a metric space and questions why it is restricted to a discrete metric space. However, it appears to be a misprint as later in the text it states that an arbitrary metric space (X,d) is said to be complete if every Cauchy sequence converges. Other readers also confirm that it is likely an error and Peter can now continue reading with confidence.
Math Amateur
Gold Member
MHB
I am reading John B. Conway's book: Ä First Course in Analysis and am focused on Chapter 5: Metric and Euclidean Spaces ... and in particular I am focused on Section 5.2: Sequences and Completeness ...

I need some help/clarification with Conway's defintion of completeness of a metric space ...

Conway's definition of a Cauchy sequence and a complete metric space read as follows ... ... View attachment 7637In the above text from Conway we read the following:

"... ... The discrete metric space $$\displaystyle (X,d)$$ is said to be complete if every Cauchy sequence converges. ... ... My question is as follows:

Why is Conway restricting this definition to a discrete metric space ... indeed is this a misprint ... ?

Surely we can say that an arbitrary metric space $$\displaystyle (X,d)$$ is said to be complete if every Cauchy sequence converges. ... ... Hope someone can help ...

Peter

NOTE: at the beginning of Section 5.2 $$\displaystyle (X,d)$$ is declared to be a given (arbitrary) metric space ... as follows:View attachment 7638

Definitely a misprint. In fact, it looks as though the word "discrete" somehow spread from Example 5.2.4(d) to Definition 5.2.3 at some point during the copy-editing stage.

Hi Peter,

Unless there is something else in the book that suggests otherwise, I think that this is indeed an error.

After all, textbook authors, like all of us, are not immune from blunders;).

castor28 said:
Hi Peter,

Unless there is something else in the book that suggests otherwise, I think that this is indeed an error.

After all, textbook authors, like all of us, are not immune from blunders;).
Thanks Opalg and castor28 ...

... now have the confidence to move on with reading the text ...

Peter

castor28 said:
Hi Peter,

Unless there is something else in the book that suggests otherwise, I think that this is indeed an error.

After all, textbook authors, like all of us, are not immune from blunders;).
I om!

## What is a complete metric space?

A complete metric space is a mathematical concept in which every Cauchy sequence, or a sequence in which the terms become arbitrarily close to each other as the sequence progresses, has a limit within the space. This means that all convergent sequences in a complete metric space have a limit that also belongs to the space.

## What is a Cauchy sequence?

A Cauchy sequence is a sequence of numbers in which the terms become arbitrarily close to each other as the sequence progresses. This means that for any small distance, there exists a point in the sequence after which all subsequent points are within that distance from each other.

## How is the completeness of a metric space determined?

The completeness of a metric space is determined by the existence of limits for all Cauchy sequences within the space. If all Cauchy sequences have a limit within the space, then the space is considered complete.

## What is the significance of completeness in a metric space?

The completeness of a metric space is important because it ensures that all convergent sequences within the space have a limit that also belongs to the space. This allows for the extension of many concepts from finite-dimensional spaces to infinite-dimensional spaces.

## How is the completeness of a metric space related to continuity?

The completeness of a metric space is closely related to continuity. In a complete metric space, a function is continuous if and only if it preserves Cauchy sequences, meaning that the limit of the function applied to a Cauchy sequence is equal to the function applied to the limit of the original sequence. This property is known as the Cauchy criterion for continuity.

• Topology and Analysis
Replies
7
Views
2K
• Topology and Analysis
Replies
5
Views
2K
• Topology and Analysis
Replies
2
Views
2K
• Topology and Analysis
Replies
1
Views
1K
• Topology and Analysis
Replies
3
Views
2K
• Topology and Analysis
Replies
2
Views
1K
• Topology and Analysis
Replies
3
Views
1K
• Topology and Analysis
Replies
6
Views
2K
• Topology and Analysis
Replies
2
Views
1K
• Topology and Analysis
Replies
7
Views
2K