Compactness (mathematics) – Wikipedia

Posted on March 28, 2017 by lordneo

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In topology, we say of a space that it is compact If it is separated and it checks the property of Borel-Lebesgue. The separation condition is sometimes omitted and certain results remain true, such as the generalized terminal theorem or the Tychonov theorem. There compactness Allows you to pass certain properties from the local to the overall, that is to say that a true property in the vicinity of each point becomes valid in a uniform manner throughout the compact.

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Several properties of segments of the real right ℝ are generalized in compact spaces, which gives them a privileged role in various fields of mathematics. In particular, they are useful to prove the existence of Extrema for a digital function.

The name of this property pays tribute to the French mathematicians Émile Borel and Henri Lebesgue, because the theorem which bears their name establishes that any segment of ℝ is compact and, more generally, that Compacts of ℝ ⁿare the limited closed .

A more intuitive approach to compactness in the particular case of metric spaces is detailed in the article “Sequential compactness”.

Preliminary definition: either

{displaystyle E}

$E$ A set and

{displaystyle A}

$A$ a part of

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{displaystyle E}

$E$ . We say that a family

{Displaystyle (u_ {i}) _ {iin i}}

$(U_{i})_{{iin I}}$ the parties de

{displaystyle E}

$E$ cover

{displaystyle A}

$A$ Si Réunion

{Displaystyle BigCup _ {Iin i} u_ {i}}}

${displaystyle bigcup _{iin I}U_{i}}$ contains

{displaystyle A}

$A$ .

Property of Borel-Lebesgue for segments: or a segment

{displaystyle [a,b]}

$[a,b]$ from the real right. From any open cover of this segment, you can extract a finished sub-covering. That is to say that for any family

{Displaystyle (u_ {i}) _ {iin i}}

$(U_{i})_{{iin I}}$ open sets covering

{displaystyle [a,b]}

$[a,b]$ , there is a finite part

{DisplayStyle J}

$J$ of

{displaystyle I}

$I$ such as the sub-family

{Displaystyle (u_ {i}) _ {Iin J}}

${displaystyle (U_{i})_{iin J}}$ covers

{displaystyle [a,b]}

$[a,b]$ .

For a demonstration of this property see the theorem of Borel-Lebesgue, also called Heine-Borel theorem.

The property of Borel-Lebesgue is closely linked to a property of the bounded consequences of real: any bounded follow-up of real, one can extract a convergent suite. The link between the two properties is explained below (in the “Bolzano-Weaierstrass and sequential compactness” section).

From either of these properties it is possible to draw some important consequences on digital functions. In particular: the image of a segment by a continuous application is not only (according to the theorem of intermediate values) an interval, but it is even a segment (theorem of the terminals), and the function is then uniformly continuous ( Heine theorem).

The property of Borel-Lebesgue (as well as the sequential compactness) can be formulated as a property intrinsic of the topological space studied (here: space

{displaystyle [a,b]}

$[a,b]$ Equipped with its usual topology), regardless of the fact that it is, possibly, included in a “larger” topological space (here:

{displaystyle mathbb {R} }

$mathbb {R}$ ) and is therefore provided with the induced topology. In this sense, the notion of “compact part” (from a topological space) fundamentally differs from that, for example, of “closed part”.

Table of Contents

Borel-Lebesgue axiom and general definition of compacts [ modifier | Modifier and code ]

A topological space

{displaystyle E}

$E$ is said quasi-compact If he checks the Borel-Lebesgue Axiome : of any open cover of

{displaystyle E}

$E$ , you can extract a finished sub-recovery. The space is said compact When it is also separated in the sense of Hausdorff (

{displaystyle T_{2}}

$T_2$ ). A part

{displaystyle K}

$K$ of AND is said (almost) compact if K Equipped with induced topology is (almost) compact.

So that

{displaystyle E}

$E$ is almost compact, it is enough that any recovery of

{displaystyle E}

$E$ By open of a fixed base has a finished sub-review.

It is even enough to be so for a prebase ( cf. Properties of the prebates, Alexander theorem).

By going to complementary, the property of Borel-Lebesgue is equivalent to: if

{displaystyle (F_{i})_{iin I}}

$(F_{i})_{{iin I}}$ is a family ^{[ first ]}closed such that

{displaystyle bigcap _{iin I}F_{i}=emptyset }

${displaystyle bigcap _{iin I}F_{i}=emptyset }$ , then we can extract a finished family (

{displaystyle (F_{i})_{iin J}}

${displaystyle (F_{i})_{iin J}}$ , with

{displaystyle Jsubset I}

${displaystyle Jsubset I}$ , such as

{displaystyle bigcap _{iin J}F_{i}=emptyset }

${displaystyle bigcap _{iin J}F_{i}=emptyset }$ . Or even, by contraposition: if

{displaystyle (F_{i})_{iin I}}

$(F_{i})_{{iin I}}$ is a family of closed, of which any subfamily finished has an uncontempt intersection, then

{displaystyle bigcap _{iin I}F_{i}}

${displaystyle bigcap _{iin I}F_{i}}$ is not empty. In an equivalent way: any non -empty family of non -empty closed stable by finished intersections has an uncontestable intersection.

A topological space

{displaystyle X}

$X$ is almost compact if ^{[ 2 ]}(and only if) the intersection of any channel ^{[ 3 ]}not empty of closed not empty of

{displaystyle X}

$X$ is not empty.

NB: In Anglo-Saxon terminology, the definition is slightly different. Unless otherwise stated, the English-speaking compact is an almost French-speaking quasi-compact (English speakers specify “compact hausdorff” if they want separation). All properties therefore do not apply in general, except under the hypothesis that space is separated.

And

{displaystyle E}

$E$ is a topological space, the following three properties are equivalent:

${displaystyle E}$
any filter on ${displaystyle E}$
all ultrafiltre on ${displaystyle E}$

Demonstration

1. ⇒ 2.
Let ℱ a filter on

{displaystyle E}

$E$ .
SO

m tume Slegles Bhaken no embone empuxant m krove m kome m kome hor h other h other horine mate halm halm halm malm malm .

${displaystyle bigcap nolimits _{Fin {text{ℱ}}}!{bar {F}}neq varnothing }$ .
Indeed the intersection of a finite number of elements of ℱ being not empty, any finite intersection

{displaystyle {overline {F_{1}}}cap ldots cap {overline {F_{n}}}}

${displaystyle {overline {F_{1}}}cap ldots cap {overline {F_{n}}}}$ is not empty (where

{displaystyle F_{1},ldots ,F_{n}in {text{ℱ}}}

${displaystyle F_{1},ldots ,F_{n}in {text{ℱ}}}$ ).
By compactness of

{displaystyle E}

$E$ The above intersection is not empty and therefore has an element

{displaystyle a}

$a$ .
This element is a member of everything

{displaystyle Fin {text{ℱ}}}

${displaystyle Fin {text{ℱ}}}$ , that is to say that any neighborhood of

{displaystyle a}

$a$ encounter

{displaystyle F}

$F$ ; It is the very definition of the adhesion of a point to a filter.

2. ⇒ 3.
Either ? an ultrafilter on

{displaystyle E}

$E$ , And

{displaystyle a}

$a$ One of his adherent points, which exists by hypothesis.
Either

{displaystyle V}

$V$ any neighborhood of

{displaystyle a}

$a$ ; By maximum? we have: or else

{Displaystyle Vin {text {?}}}}

${displaystyle Vin {text{?}}}$ or

{displaystyle Esmallsetminus Vin {text{?}}}

${displaystyle Esmallsetminus Vin {text{?}}}$ . If this last assertion was verified, the fact that

{displaystyle Vcap (Esetminus V)=varnothing }

${displaystyle Vcap (Esetminus V)=varnothing }$ would contradict the grip of

{displaystyle a}

$a$ To ?. It is then

{displaystyle V}

$V$ Who is an element of?, So? converge

{displaystyle a}

$a$ .

3. ⇒ 1.
Consider

{displaystyle (F_{i})_{iin I}}

$(F_{i})_{{iin I}}$ , a family of closed

{displaystyle E}

$E$ whose intersection of any finished subfamily is not empty.
The set of intersections

{displaystyle bigcap nolimits _{1leqslant kleqslant n}!F_{i_{k}}}

${displaystyle bigcap nolimits _{1leqslant kleqslant n}!F_{i_{k}}}$ (Or

{displaystyle i_{1},ldots ,i_{n}in I}

${displaystyle i_{1},ldots ,i_{n}in I}$ ) therefore constitutes a filter base, which generates a filter which is itself included in an ultrafiltre ?.
By hypothesis? converge towards a point

{displaystyle a}

$a$ . Either

{displaystyle iin I}

$iin I$ , and either

{displaystyle V}

$V$ any neighborhood of

{displaystyle a}

$a$ , SO

{displaystyle Vcap F_{i}neq varnothing }

${displaystyle Vcap F_{i}neq varnothing }$ Because these two sets are elements of? ; we deduce that

{displaystyle ain {overline {F_{i}}}=F_{i}}

${displaystyle ain {overline {F_{i}}}=F_{i}}$ .
It follows that

{displaystyle ain bigcap nolimits _{iin I}!F_{i}}

${displaystyle ain bigcap nolimits _{iin I}!F_{i}}$ ; The latter set is therefore unaccompanied and the property of Borel-Lebesgue (version in terms of closed) is satisfied for

{displaystyle E}

$E$ .

A separate topological space is compact if and only if any generalized suite has at least one adhesion value, in other words a convergent generalized subsidiary. This equivalent definition is rarely used. It is particularly adequate to prove that any compact product is compact.

In any quasi-compact space, a filter which has only an adherent point converges towards this point; In a compact space therefore separated, this sufficient condition of convergence is obviously necessary ^{[ 4 ]}.

Any finished space is almost compact since it has only a finite number of open.
In a separate space, given a convergent suite, the whole made up of the terms of the suite as well as the limit is compact. Indeed, from any open cover, one can extract an open containing the limit; As there is only a finite number of terms out of this open, it is easy to find a finished sub-recovery.
Any whole equipped with cofinie topology is almost compact.
The set of cantor is compact, as closed from the compact [0, 1].
The Hilbert cube is compact, as a product of a (countable) family of compact copies [0, 1].

Compact and closed [ modifier | Modifier and code ]

In a separate space, two compact disjointes are always included in two disjoined open ^{[ 5 ]}.
Any compact part of a separate space is closed ^{[ 6 ]}.
Any part closed of a compact space is (almost) compact ^{[ 6 ]}.

It is easily deduced from the two previous properties than in a separate space, any intersection of a family not empty of compacts is compact ^{[ 7 ]}.

In a quasi-compact space, the intersection of any decreasing following of non-empty closed is not empty ^{[ 8 ]}, SO :

‘ nested compact theorem »: In any topological space, the intersection of any decreasing result of non -empty compacts is (a compact) not empty
(Considering these compacts as closed from the first of them).

N. B .: Most of these properties do not extend in the unrevised case.

Counterexample

In the pair {0, 1} fitted with the coarse topology, {0} and {1} are compact but not closed;
In the product of this pair by ℝ (provided with its usual topology), {1} × [–1, 1] and ({0} × [–1, 0]) ∪ ({1} ×] 0, 1] ) are canonically homeomorphic to [–1, 1] therefore compact (but not closed) and their intersection, {1} ×] 0, 1], is not even almost compact;
This same example shows that compactness is not preserved by finished meetings (only the quasi-compactness is).

This allows you to refine the theorem of nested compacts:

Any intersection of a decreasing suite of related compacts is related.

Demonstration of these two properties

A compact space is normal : in a compact space AND , are A , B two disjoined closed. According to the above properties, A And B are then two disjoined compacts and AND is separated, so that A And B are included in two disjoint open, which proves that AND is normal.
Any intersection of a decreasing suite of related compacts is related : either ( F _n) a decreasing suite of compacts; Suppose his intersection K is not related. There are therefore two closed F And G of F ₀such as F ⋂ K And G ⋂ K are not empty and complementary in K . By normality of F ₀, there are then two disjoint open IN And IN of F ₀which contain respectively F ⋂ K And G ⋂ K . Note IN the meeting of these two open and G _nthe closed F _nIN . THE G _nform a decreasing sequence of compacts of empty intersection, therefore one of the G _nis empty. THE F _ncorrespondent is then included in the disjoined meeting W = u ⋃ IN . Like this F _nMeet each of the two open IN And IN , it is not related, which demonstrates the contradiction of the proposal.

Other properties [ modifier | Modifier and code ]

A real normally standardized vector space is of finite dimension if and only if its compacts are its limited closed.

The Cartesian compact product, provided with product topology, is compact.

More specifically: any quasi-compact product is almost compact; This result, known as Tykhonov’s theorem, is equivalent to the axiom of choice.

Any discreet and closed part of a quasi-compact is over.

Kuratowski theorem ^{[ 9 ]}-Ant ^{[ ten ]}: A separate space X is compact if and only if for any space AND , projection p _AND: X × AND → AND is a closed application.

More generally, a space X is almost compact if and only if it checks this property ^{[ 11 ]}.

Demonstration

And X is almost compacting so the projection p _AND: X × AND → AND is closed for everything AND : See “Lemma of the tube”.
Conversely, suppose that the projection p _AND: X × AND → AND is closed for everything AND And let’s show that X is almost compact. Either ( IN _i) _{i ∈ I}an open cover of X . Note
- ∞ an arbitrary element not belonging to X ,
- AND the disjoint union of X a you singleton {☀},
- AND The set of parts of X which are meetings of a finite number of IN _i,
- B The set of parts of AND who or do not contain ∞, or are complementary in AND an element of AND ,
- ∆ to diagonals of X ( i.e. The set of couples ( x , x ) When x go through X ), which is a part of X × X therefore X × AND .

We easily check that B constitutes the basis of a topology on AND And we apply the hypothesis to this topological space AND : the image by p _ANDclosed ∆ is a closed of AND . Moreover, p _AND( ∆ ) (which contains X ) does not contain ∞ (because X × {∞} is included in a disjoined open of ∆: the meeting of IN _i× ( AND IN _i)). This proves that {∞} is an open of AND . We deduce that X belongs to AND , what concludes.

As a result, any application of closed graph of any space in a quasi-compact space is continuous ^{[ twelfth ]}.

Either f : A → B with B quasicompact et Gr( f ) closed in A × B , and either F a closed of B . SO f ⁻¹( F ) is a closed of A , as an image of the closed ( A × F ) ∩GR ( f ) by the closed application p _A: A × B → A .

Compactness and continuity [ modifier | Modifier and code ]

The image of a compact, by a continuous application to values in a separate space, is compact ^{[ 14 ]}.
This property allows you to exhibit global extrema for continuous functions with real values. Here are some examples:
- Theorem of terminals (or Weierstrass theorem): “Any continuous application of a real segment in ℝ is limited and reaches its terminals” (attached to the theorem of intermediate values, it ensures that this image is in fact a segment);
- Fermat point problem. An ABC triangle given, it is asked to prove that there is a point M such as the sum of distances AM + BM + CM or minimal. We first notice that it is useless to seek M too far from the points A,B,C . Consideration of continuous application M ↦ AM + BM + CM On a sufficiently large closed radius of radius allows you to apply the theorem: there is a global minimum. This observation can serve as a starting point for an explicit construction;
- Distance from a point to a closed of ℝ ⁿ. Be F a closed part not empty of ℝ ⁿAnd x A point of ℝ ⁿ. It is a question of proving that there is a point f of F closer to x that all the others. Again, it is useless to seek f too far from x . We can therefore limit ourselves to the intersection of F and a closed ball, which constitutes a compact according to the theorem of Borel-Lebesgue, and introduce the distance function to x , which is continuous;

A continuous values function on a compact always reaches its maximum.

Isoperimetric character of a regular polygon, a question open since Antiquity. The object is to know what is the polygon at n Sides which has the largest area, for a given perimeter. Quite simple geometric reasoning show that the only possible candidate is the regular polygon, a result demonstrated since Greek antiquity. On the other hand, the existence of a solution to this question remained open until XIX ^{It is}century.
To understand the nature of the demonstration, the easiest way is to consider the case of the triangle, illustrated on the right figure. The triangles considered are all of perimeter 3, they are identified with a couple ( c , φ) where c designates the length on one side and φ the angle between two sides, one of which is the length c . Function f is the one who, with a couple, combines the surface of the triangle. It is only necessary to study the area where c is between 0 and ³⁄ ₂and φ between 0 and π. This area is a compact of ℝ ². L’application f is continuous, it therefore reaches its maximum, in this case to the point (1, ^Pi⁄ ₃). The existence of this maximum was the “missing link” for a complete demonstration.
For the triangle, a little analysis allows just as well to demonstrate the result. For the general case of polygon at n Sides, it is not very difficult to build a demonstration similar to that presented here, thanks to the concept of compact. The analytical solution is however really heavy. A detailed demonstration is presented in the article “isoperimetric theorem”.

A corollary of the theorem on the continuous image of a compact is:
Any continuous application of a compact space in a separate space is closed. In particular, if it is bijective then it is a homeomorphism.
Links between compact and closed, we immediately deduce that such an application is even clean.
For any continuous application f of a compact metric space X In a separate space, the compact f ( X ) is mild (for example: the image of any path in a separate space is mighty). Thanks to a general characterization of the dilters of the image of a metric space by a closed continuous application ^{[ 15 ]}^,^{[ 16 ]}, we even have equivalence: a metric space X is compact if and only if all its separate images are bound.

Bolzano-Weaierstrass theorem and sequential compactness [ modifier | Modifier and code ]

In a compact space, any infinite part has at least one limit point. More generally, any space X quasi-compact is countingly compact, that is to say that any infinite part of X has at least one point of accumulation or that in X , any following has at least one grip value ^{[ 17 ]}. The reciprocal is generally false, but true if the space is mild: when K is a small space (automatically separated), the Bolzano-Weaierstrass theorem states that K is compact if and only if it is sequentially compact, that is to say if, in K , any following has a convergent subsidiary.

The first non -countable ordinal (equipped with the topology of the order) and the long right are sequentially compact but not compact (they are however locally compact). Conversely, the product space [0, 1] ^ℝ(that is to say the space of the applications of ℝ in [0, 1], equipped with the topology of simple convergence) and the compactified of Stone-Sch of ℕ (that is to say the spectrum algebra $ℓ \infty$ Bounded) are compact but not sequentially compact. These four spaces are therefore enumerably compact and not bound.

↑ If we do not specify “family does not see », It must be agreed that in this context, the intersection of a family empty of parts of a space ${displaystyle X}$
↑ (in) Günter Bruns , ‘ A lemma on directed sets and chains » , Archives of mathematics , vol. 18, n ^O6, 1967 , p. 561-563 ( read online ) .
↑ A chain of games of ${displaystyle X}$
↑ Bourbaki, TG I.60, Gustave Choquet, Analysis lessons, volume II: topology , p. 35 And Hervé Queffélec, Topology , Dunod, 2007 , 3 ^{It is} ed. , p. 70 .
↑ For a demonstration (using a generalization of the lemma of the tube), see for example This corrected exercise on the Wikiverity .
↑ ^{a et b}For a demonstration, see for example Jacques Dixmier, General topology , PUF, 1981, 4.2.6 and 4.2.7, p. 53 , or the course Compact: First properties on the Wikiverity .
↑ For a demonstration, see for example the course Compact: First properties on the Wikiverity .
↑ In other words: everything almost compact is countingly compact.
↑ Casimir Kuratowski, « Evaluation of the Borélian or projective class of a set of points using logical symbols », Foundations of mathematics , vol. 17, n ^O1, 1931 , p. 249-272 ( read online ) .
↑ (in) S. Mrówka, ‘ Compactness and product spaces » , Interview mathematics , vol. 7, n ^O1, 1959 , p. 19-22 ( read online ) .
↑ (in) M. M. Choban , « Closed maps » , in K. P. Hart, J.-I. Nagata and J. E. Vaughan, Encyclopedia of General Topology , Elsevier, 2004 (ISBN 978-0-44450355-8 , read online ) , p. 89 (by translating English compact by our quasi-compact ).
↑ (in) James Munkres, Topology , Prentice Hall, 2000 , 2 ^{It is} ed. ( read online ) , p. 171 .
↑ This proof is extended to multifunction in the article “Hemvétinity”.
↑ For a demonstration, see for example the course Compactness and continuous applications on the Wikiverity .
↑ (in) Stephen Willard , ‘ Metric spaces all of whose decompositions are metric » , Proc. Amer. Math. Soc. , vol. 21, 1969 , p. 126-128 ( read online ) .
↑ (in) Thanks Morita and sitiro Feed , ‘ Closed mappings and metric spaces » , Proc. Japan Acad. , vol. 32, n ^O1, 1956 , p. 10-14 ( read online ) .
↑ We deduce that in such a space, any continuation which has only a value of adhesion converges towards this value.

N. Bourbaki , Elements of mathematics, book III: general topology [Detail of editions] , chapter I
Jean-Paul Pier « Genesis and evolution of the idea of compact », Revue d’histoire des sciences and their applications , vol. 14, n ^O2, 1961 , p. 169-179 ( read online )
Jean Paul Pier « History of the concept of compactness », Mathematical History , vol. 7, n ^O4, November 1980 , p. 425-443 (DOI 10.1016/0315-0860 (80) 90006-3 )
Georges Skandalis, Topology and analysis 3 ^{It is}year , Dunod, coll. “SUP Sciences”, 2001
Claude Wagschal, Topology and functional analysis , Hermann, coll. “Methods”, 1995

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Compactness (mathematics) – Wikipedia

Borel-Lebesgue axiom and general definition of compacts [ modifier | Modifier and code ]

Compact and closed [ modifier | Modifier and code ]

Other properties [ modifier | Modifier and code ]

Compactness and continuity [ modifier | Modifier and code ]

Bolzano-Weaierstrass theorem and sequential compactness [ modifier | Modifier and code ]

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