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Final Answers
© 2000-2015   Gérard P. Michon, Ph.D.

Completeness  &
Uniform  Spaces

 Augustin Cauchy 
 1789-1857  Stefan Banach 
 1892-1945
 You can't do anything with
 a space that's not complete
.
Laurent Schwartz  (1915-2002)
lecturing in the Fall of 1977.

Related articles on this site:

Related Links  (Outside this Site)

Cauchy Sequences and Complete Metric Spaces  by  Mark Walker  (Econ 519)
Complete Metric Spaces  by  Lewkeeratiyutkul Wicharn  (Chula)
 
Wikipedia :     Complete metric space   |   Uniform spaces   |   Dedekind cut   |   Dedekind-MacNeille completion

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Completeness


 Augustin Cauchy 
 1789-1857 (2007-11-15)   Completeness in a Metric Space
metric  concept which is  not  a topological property.

metric space  is  complete  when any Cauchy sequence in it converges.

Completeness is fundamentally a  metric  property  (the definition of completeness depends critically on the definition of a distance).  Even if two distances are defined on the same set which induce the  same topology  on that space, it's quite possible that one distance defines a  complete  space and the other one doesn't.

A metric space is compact if and only if it's  complete  and  totally bounded.

A subset of a metric space is said to be totally bounded when it can be covered by finitely many balls of radius  r,  for any given radius  r.  In a Euclidean space of infinitely many dimensions, a bounded set  (like a ball of unit radius)  need not be totally bounded.  Actually, a closed ball is compact only in a space of finitely many dimensions.

By definition, a  topological property  is preserved by any homeomorphism.  This is  not  always the case for completeness.  For example, R  is complete and it's homeomorphic to the open interval  ]0,1[  which is  not.  (HINT :  A positive sequence that tends to  0  in [0,1]  isn't convergent in  ]0,1[ .)

metrizable  space is defined as a topological space homeomorphic to a metric space.  Such a space is called  complete-metrizable  when at least one metric space homeomorphic to it is complete.  That's a topological property  (since it's clearly preserved by homeomorphisms)  but it's difficult to characterize in practice.


(2014-12-05)   Completeness in a Uniform Space   (Weil, 1937)
Completeness can also be defined in  uniform  topological spaces.

uniform space  is  complete  when every  Cauchy filter  in it converges.

Motivation :

What made it possible to define completeness in a metric space is the existence of a family of relations  (i.e., subsets of the cartesian product)  dependent on a single positive parameter  a:

Ua  =   { (x,y)  |  d(x,y) < a }

The triangular inequality for the distance  d  enables us to construct a relation  V = Ua/  which is. loosely speaking,  at most  half as wide  as the relation  U = Ua .  The crucial aspect can be expressed as follows, in terms of the  composition  of relations  (this simple exercise is left to the reader).

V o V   Í   U

This expression no longer involves any explicit reference to distances.  The postulated existence of a sequence of relations based on this composition pattern will enable us to generalize the notion of Cauchy sequences and completeness without using the notion of a distance...

 Come back later, we're
 still working on this one...

nLab :   Uniform spaces     Wikipedia :   Uniform spaces   |   Filter (1937, Henri Cartan)


(2014-12-05)   Baire Category Theorem   (Baire, 1899)
In a complete space, countable intersections of dense open sets are dense.

This proposition is a theorem in ZFC.  In ZF, it turns out to be equivalent to the Axiom of dependent choice (DC), a weak form of the full Axiom of choice  which is sufficient for conducting real analysis without implying the repugnant existence of non-measurable sets of reals

For that reason,  Henri Garnir (1921-1985)  has proposed to adopt DC or, equivalently, the Baire Category theorem instead of AC among the axioms of Set theory.  This allowed him to postulate that every set of reals is almost open, which makes every set of reals Lebesgue-measurable and dismisses the Banach-Tarski paradox.

 Come back later, we're
 still working on this one...

Baire space   |   Baire category theorem


(2007-11-15)   Completeness Redux
Tentative  (flawed)  topological characterizations of completeness.

Let's try topological characterizations of completeness to see how such attempts fail.  For example, let's examine the following property:

Any decreasing sequence of nonempty closed sets has a nonempty intersection:

{   " i Î NAi ¹ Æ  is closed,   Ai+1 Í Ai   }     Þ     Æ ¹  Ç   Ai
  i Î N  

This would seem like a good candidate for a topological characterization of completeness until you realize that it's not even true for a noncompact complete space like  R  in which there are indeed nested collection of nonempty closed sets with an empty intersection.  One example is  Ai = [i,¥[.

For families of  compact  closed sets, the above characterization still fails for metric spaces of infinitely many dimensions (where closed balls are not compact).

All told, a topological space can only be said to be complete  with respect to  a specific distance compatible with its topology  (two different distances may induce the same topology but the space can be complete with respect to one metric and not the other).  Such a space is called either  topologically complete  or  complete-metrizable.  There is simply no easy way to characterize that property...

Fréchet space   =   Locally convex vector space, complete with respect to a translation-invariant metric.
Cantor's intersection theorem


 Stefan Banach 
 1892-1945 (2007-11-06)   Banach Spaces   (1920)
Banach Spaces are  complete  normed vector spaces.

Banach space  is a  normed vector space  which is complete  (which is to say that every  Cauchy sequence  in it converges).  The concept is named after the Polish mathematician  Stefan Banach (1892-1945) who axiomatized the idea in his doctoral dissertation  (1920)  and made it popular through his 1931 foundational book on  functional analysis,  which was translated in French the next year  (Théorie des opérations linéaires,  1932).

Arguably, Banach spaces are the main backdrop for modern  analysis, the branch of mathematics which revolves around the very notion of  limit  (it would be hazardous to discuss limits in a space that's not  complete).

The key example which motivated Stefan Banach :

The  Riesz-Fischer theorem  (1907)  states that  Lp  is a Banach space.

Lp spaces and Banach spaces   |   Espaces de Lebesgue  Lp   |   Riesz-Fischer theorem (1907)
Ernst Fischer (1875-1954)   |   Frigyes Riesz (1880-1956)   |   Stefan Banach (1892-1945)


 Maurice Frechet 
 1878-1973 (2013-01-22)   Fréchet Spaces
The key properties of Banach spaces for distances not based on a norm.

In 1906, Maurice Fréchet had proposed the general notion of a metric space without assuming an underlying vector structure.

He realized that the key results that make Banach spaces interesting could also be obtained for vector spaces that are complete with respect to a distance  not  associated with a norm.  He thus investigated structures more general than Banach spaces, which are now called  Fréchet spaces :

Fréchet space  is a locally-convex vector space which is complete with respect to a given translation-invariant metric.

Wikipedia :   Locally convex vector space   |   Fréchet space

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