Talk:Complete metric space

message: from this article i don't really understand what is the difference between a closed set and a complete space.
maybe someone can give an example of a closed space which ins't complete? —The preceding unsigned comment was added by 217.132.104.254 (talk) 09:38, 8 February 2007 (UTC)
thread_title:  complete spaces vs. closed set 

message: Is there a reason this specific formulation of the Completeness axiom was chosen over, say, the Supremum formulation, or that of Monotonic Convergence? Should these not at least be mentioned with a discussion of the benefits / downfalls of different formulations?
I ask this specifically as the Supremum (or Infimum) formulation is often easier for the non-mathematician to understand. In a sense the most complex formulation is that represented here, and the question is whether this is where Wikipedia articles should be aimed.
-Jdr
[13/3/2007 - 10:06pm GMT]
:: "Supremum" makes sense when there is an ordering, but in metric spaces generally there is no supremum.  Which "monotic convergence" formultation are you talking about?  Does it work in metric spaces generally?  "Monotonic" seems to connote an ordering. Michael Hardy 22:54, 13 March 2007 (UTC)
thread_title:  Other formulations of the completeness axiom 

message: In the text there is the example
"The topological vector space R^ω of sequences of real numbers which have finitely many nonzero terms (the topology of this space can alternatively be defined as the limit topology of the Rn or as the coproduct of infinitely many copies of R) is not complete (even though its underlying field is). The completion of this space is the product ΠR of infinitely many copies of R. If instead we endow the space with the lp norm, its completion is the space lp(N)."
This contains many false statements, like
1.) The beginning
"The topological vector space R^ω of sequences of real numbers which have finitely many nonzero terms"
is already a bit strange, since so far no topology is given and therefore R^ω is just a vector space and not yet a topological one. There are many totally different topologies on this vector space.
2.) "the topology of this space can alternatively be defined as the limit topology of the Rn or as the coproduct of infinitely many copies of R"
Yes, it is possible to endow this vector space with this particular topology, but then is not metrizable (and therefore not at all a good example for a "Complete Metric Space").
By the way, the space R^ω endowed with this topology is complete but not with respect to a metric (since there is none defining this topology) but with respect to the uniform strucutre as a topological vector space.
3.) The completion of this coproduct in number (2.) is NOT the product ΠR, since the coproduct is already complete as mentioned above.
It is true that the product can be the completion of R^ω, but with another topology (which that in fact will be again metrizable)
4.) I admit that the last sentence is correct. "If instead we endow the space with the lp norm, its completion is the space lp(N)."
I would suggest that we remove this example or at least restrict it to the case where we put an lp-norm on it.
--130.83.2.27 08:09, 7 August 2007 (UTC)
replies: text: I cut out the example per your item 1 (I am not competent enough to judge items 2 and 3. :) There are many good examples in the text that we can get by without this very technical example, especially that it is not well-written and possibly incorrect. Oleg Alexandrov (talk) 19:43, 7 August 2007 (UTC)
thread_title:  Strange Example 

message: I think the following fact (from A. A. Kolmogorov, S. V. Fomin "Introductory real analysis") should be added in the section "Some Theorems" (or even under a new title „A Completeness Criterion of a Metric Space“) or in the as of now poor article "Nested intervals":
Page 60 THEOREM 2 (Nested sphere theorem). A metric space R is complete if 
and only if every nested sequence  \{S_n\} = \{S[x_n, r_n]\}  of closed spheres in R such that  r_n \rightarrow 0  as n \rightarrow \infty  has a nonempty intersection.
Matumba (talk) 00:06, 23 February 2009 (UTC)
replies: text: I think this has to do with expansion constants, so I added a bit. -- Taku (talk) 21:57, 23 February 2009 (UTC)
thread_title:  Characterization of complete spaces through nested balls 

message: Who uses this term for complete metric spaces? Usually it refers to something completely different. --Chricho ∀ (talk) 16:01, 16 June 2014 (UTC)
thread_title:  “Cauchy space” 

message: :It has the following universal property: if N is any complete metric space and f is any uniformly continuous function from M to N, then there exists a unique uniformly continuous function f′ from M′ to N, which extends f.  The space M is determined up to isometry by this property, and is called the completion of M.
This property has any space M′ which contains M  as a dense subspace. So, what means that this property determinates M′ up to isometry?
109.105.162.103 (talk) 06:24, 22 February 2019 (UTC)
thread_title: Universal Property of Completion