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

[Math Amateur]

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

 

[Opalg]

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.

 

[castor28]

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;).

 

[Math Amateur]

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

 

[HallsofIvy]

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!