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(2007-11-15)  
A metric  concept which is not  a topological property.

A metric space  is complete  when anyCauchysequence in it converges.

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

Ametric space iscompact if and only if it's complete  and totally bounded.

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By definition, a topological property  is preservedby anyhomeomorphism. This is not  always the case for completeness. For example,  is complete and it's homeomorphic tothe openinterval  ]0,1[ which is not. (:  A positive sequence that tends to 0  in [0,1]  isn't convergent in  ]0,1[ .)

A metrizable  space is defined asa topological space homeomorphic to a metric space. Such a space is called complete-metrizable  when at least onemetric 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.

(2021-08-11)  .  uniform continuity
Uniformly continuous functions respect completeness.

Recall that the following definitions hold for a function f  which mapsone  metric space (X₁,d₁)  into another  (X₂,d₂) :

 f  is continuous when:

> 0 ,  xX₁ ,  > 0 ,  d₁(x,y) ≤ d₂(f (x),f (y) ) ≤

 f  is uniformly  continuous when:

> 0 ,  > 0 ,  xX₁ ,  d₁(x,y) ≤ d₂(f (x),f (y) ) ≤

The order of the quantifiers matters: In the first case,    can depend on  x. In the second case,  it cannot.

If X₁  is complete,  so is f (X₁) when f  is uniformly continuous.
That may not be the case if f  is merely continuous.

Cauchy-regular functions :

A function is said to be Cauchy-regular  (or Cauchy-continuous) if it transforms any Cauchy sequence  into another Cauchy sequence. Uniformly continuous  functions are Cauchy-regular.

Heine-Cantor theorem,  for metric spaces :

Theorem :   Continuity on a compact set  is always  uniform.

Proof :   To establish that in the case of metric spaces  (where uniform continuity is defined as a above)  let's consider any continuous function f.

For any  > 0,  the continuity of f implies that,  for any given  x,  there's a quantity _x  such that:

d₁(x,y) ≤_xd₂(f (x),f (y) ) ≤ ½

To any  x  in X₁  we associate a particular open set:

U _x  =  { y : d₁(x,y)  <  ½_x }

The family formed by all of these is an open cover  of X₁ (:   x U_x ). As X₁  is assumed to be compact. we can extract from that family a finite  subcover,  for which we use the folowwing notation:

U _xi   =   { y : d₁(x_i,y)  < ½ _xi }    with  i = 1,2,3,4 ... n

Uniform Continuity and Derivatives :

If f  is a real function of a real variable defined on the interval A  and differentiablein the interior Å  of A, then f  is uniformly continuous  on  A iff  its derivative f ' is bounded on  Å.

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

Topological structures  can be too permissive while the metric structuresof normed spaces can be too strict a requirement. Uniform spaces seem just right to capture essential fruitful aspects of space. Uniform spaces are to uniform continuity what topological spaces are to ordinary continuity.

A 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 afamily ofrelations  (i.e., subsets of the cartesian product) dependent on a single positive parameter a:

U_a  =   { (x,y)  |  d(x,y) <a }

Thetriangular inequality for the distance  d  enables us toconstruct a relation  V = U_a/2   which is. loosely speaking, at most half as wide  as the relation  U = U_a . The crucial aspect can be expressed as follows, in terms ofthe composition  of relations (this simple exercise is left to the reader).

V V     U

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

Filters and Ultrafilters  (Henri Cartan, 1937)

A subset F  of a poset  (P, ≤) is a filter  whenit's a nonempty downward-directed upperset,  which is to say:

• F 
• xF ,  yF ,  zF , z ≤ x ,  z ≤ y
• xF ,  yP , (x ≤ y)     (yF)

P  is always a filter of itself.  The other  filters of P are called proper  filters. An ultrafilter  is a maximal  proper filter.

(2023-10-12)    
Indexed by an arbitrary directed set, whose elements are called indices.

(2023-10-12)  
Completeness can be defined in a quasi-uniform space.

A quasi-uniform space is complete  when every Cauchy filter in it converges to a point in the space.

(2014-12-05)    (Baire, 1899)
In a complete space, countable intersections ofdenseopen sets are dense.

This proposition is a theorem inZFC. In ZF, it turns out to be equivalent to theAxiom of dependent choice (DC),a weak form of the fullAxiom of choice which is sufficient for conducting real analysis without implying therepugnantexistence of non-measurable sets of reals. 

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

Let's try topological characterizationsof 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 , Ai is closed,  Ai+1Ai		Ai
	i

This would seem like a good candidate for a topological characterization of completenessuntil you realize that it's not even true for a noncompact complete space like   in which there are indeednested collection of nonempty closed sets with an empty intersection. Example: A_i = [i,[.

For families of compact  closed sets,the above characterization still fails for metric spaces of infinitelymany 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 completewith 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 characterizethat property...

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

A Banach space  is a normed vector space  which iscomplete (which is to say that every Cauchy sequence in it converges). The concept is named after the Polish mathematician Stefan Banach (1892-1945) whoaxiomatized the idea in his doctoral dissertation  (1920) and made it popular through his 1931 foundational book on functional analysis,  whichwas 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 L^p  is a Banach space.

(2013-01-22)  
The key properties ofBanach spaces for distances not based on a norm.

In 1906,MauriceFréchet  had proposed the general notion of ametric spacewithout an underlying vector structure.

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

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