Complete space or completemetric spaceIs a space with the following properties:Cauchy sequenceAll converge to thespaceWithin.In terms of finite dimensional spacenormamount toModule of vectorLength of.But there is also a very important concept in the finite dimensional Euclidean space - the angle between vectors, especially the orthogonality of two vectors.Inner product spaceIt's specialLinear normed spaceIn this kind of space, the concept of orthogonality and projection can be introduced, and the corresponding geometry can be established in the inner product space.The distance is defined by the norm derived from the inner product,BanachSpace becomesHilbert space。[1]
In mathematics and related fields, an object hasCompleteness, that is, it does not need to add any other elements. This object can also be calledCompleteorcomplete。More precisely, this definition can be described from many different perspectives, and can be introducedCompletionThis concept.However, in different fields, "completeness" has different meanings. Especially in some fields, the process of "completion" is not called "completion". For other expressions, please refer toAlgebraic closed field、Compactification(compactification）OrGodel Incomplete Theorem 。
Intuitively speaking, one spacecompleteIt means "no hole" and "no skin", both of which are "no defect".No hole means no defect inside, and no defect means no defect on the boundary.At this point, a spacecompleteSameaggregateOfclosureIt is similar.This similarity is also reflected in the following theorem:Closed subsetIt is complete.[1]
Completion
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definition
For any metric spaceM, we can construct the correspondingComplete metric spaceM'(or expressed as
）The original metric space becomes the dense subspace of the new complete metric space.M'It has the following universal properties: ifNIs any complete metric space,fFor any slaveMreachNIf there is a uniformly continuous function ofM'reachNUniformly continuous function off'Make the functionfExtension of.A newly constructed complete metric spaceM'stayIsometric isomorphismThe meaning is uniquely determined by this property, which is calledMOfComplete space。[2]
The above definition is based onMyesM'The concept of dense subspace.We can alsoComplete spaceDefined as containingMMinimum ofComplete metric space。It can be proved that the complete space defined in this way exists, is unique (in the sense of isometric isomorphism), and is equivalent to the above definition.
For a commutative ring and itsmodel, you can also defineidealThe completeness and completeness of.See item completion for details（Ring theory)。
structure
Similar to the method of defining irrational numbers from the rational number field, we can useCauchy sequenceAdd elements to the original space to make it complete.
yesMAny two Cauchy sequences inx=(xn）Andy=(yn）, we can define the distance between them: d（x,y) = limnd(xn,yn）(The real number field is complete, so the limit exists).The metric defined in this way is only a pseudo metric, because different Cauchy sequences can converge to 0.But we can do as we do in many cases (for example, fromLreach
）, the new metric space is defined asEquivalence classWhere the equivalence class is based on therelationship(It is easy to verify that the relationship isequivalence relation ）。In this wayξx= {yyesMCauchy sequence on:
，M'={ξx：x ∈ M}, original spaceMSoxξxMapping method ofembedTo a new complete metric spaceM'Medium.Easy to verify,MIsometric isomorphic toM'Dense subspace of.
The Cantor method is a special case of the completion method: the real number field is the completion space of the rational number field as a metric space with the usual absolute value of difference as the distance.
nature
cantor Ofreal numberConstruction is a special case of the above structure;In this case, the set of real numbers can be expressed as the completion of the set of rational numbers to the absolute value.If other absolute values are taken on the set of rational numbers, the resulting complete space isP progressions。
any compactMetric spaces are complete.In fact, a metric space is compact if and only if the space is complete and completely bounded.
Any subspace of a complete space is complete if and only if it is a closed subset.
ifXIs a collection,MIs a complete metric space, then allXMap toMOfBounded functionfSet B of（X,M）Is a complete metric space, where set B（X,M）The distance in is defined as:
。
ifXIs a topological space,MIs a complete metric space, then allXMap toMOfcontinuityBounded functionfSet C ofb(X,M）Yes B（X,M）(as defined in the previous entry), so it is also complete.
（3）Open interval(0,1) is not complete.Sequence (1/2, 1/3, 1/4, 1/5,...) is Cauchy sequence but it does not converge to any point in (0, 1).
(4) OrderSIs any set,SbySAll sequences in.The following definitionsSAny two sequences on（xn）And（yn）Distance: if there is a minimum N, make
, then the defined distance is 1/N;Otherwise (all corresponding items are equal) the distance is 0.The metric space defined in this way is complete.This spaceHomeomorphismtoDiscrete spaceSCountable copies ofproduct。
Related concepts
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Completeness and closeness: previously, completeness is similar to closeness. What is the difference between "completeness" and "closeness"?The difference between them is that completeness is the property of space or set, while closure issubsetThe nature of.Usually we say that a set isClosed setorOpen set, which actually means that the set isROr a closed subset or an open subset of a topological space.For example, an open interval (0, 1) is a complete set (0, 1) or
Because the derived set of (0, 1) in these two sets is itself.But (0, 1) isRAn open subset of.Closed subset can be defined by convergence sequence, because the limit point of convergence sequence is always in the whole set, and whether the limit point is in the subset determines whether the subset is a closed subset.In contrast, there is no concept of complete set in the definition of completeness, which is why Cauchy sequence must be used instead of convergence sequence in its definition, because there must be a limit point in the definition of convergence sequence. If the limit point is not in the metric space, the distance from the point in the convergence sequence to the limit point is undefined.
Topological complete space
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Note that completeness is a metric attribute, not a topological attribute, which means that a complete metric space can be isomorphic with a non complete metric space.Real numbers are given by real numbers. They are complete, but isomorphic to the open interval (0,1). They are incomplete.
staytopologyThe space has at least one complete metric that results in a given topology, taking into account spaces that fully meet the criteria.A fully accommodative space can be characterized as a space that can be written as the intersection of a large number of open subsets of some complete metric spaces.Since the conclusion of Baire class theorem is purely topological, it also applies to these spaces.
A fully tolerable space is often referred to as topological integrity.However, the latter term is somewhat arbitrary, because measurement is not the most common structure in the topological space where integrity can be discussed (see Substitution and Overview).In fact, some authors use the term of topological integrity to describe a broader topological space, which can be completely unified.
Topological spaces isomorphic to separable holometric spaces are calledPolish space。
Substitution and generalization
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becauseCauchy sequenceIt can also be defined in a general topology group. The alternative to relying on metric structures to define integrity and build space completion is to use group structures.This is usually done inTopological vector spaceYou can see it in, but you only need to have a continuous "subtraction" operation.
The integrity definition of Cauchy sequence can also be replaced by Cauchy network or Cauchy filter.If every Cauchy net (or equivalent to every Cauchy filter) has a limit in X, then X is called complete.Cauchy space is the most common case for Cauchy net.
