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

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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 $$(X,d)$$ 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 $$(X,d)$$ is said to be complete if every Cauchy sequence converges. ... ... Hope someone can help ...

Peter

NOTE: at the beginning of Section 5.2 $$(X,d)$$ is declared to be a given (arbitrary) metric space ... as follows:View attachment 7638
 
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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!
 
A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.
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