Misplaced Pages

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100-473: Rocks of the complex include foliatied amphibolite , orthogneiss , serpentinite and lesser amounts of quartz -rich mica schist . The rocks of complex have experienced continuous exhuming during the last 15 million years as part of the ongoing Andean orogeny , with an apparent exhumation spurt 11 to 7 million years ago. This Chile location article is a stub . You can help Misplaced Pages by expanding it . This metamorphic rock -related article

200-476: A U {\displaystyle {\mathcal {U}}} -plaque with w ∈ P . Then P ∩ Q is an open neighborhood of w in Q and P ∩ Q ⊂ L ∩ Q . Since w ∈ L ∩ Q is arbitrary, it follows that L ∩ Q is open in Q . Since L is an arbitrary leaf, it follows that Q decomposes into disjoint open subsets, each of which is the intersection of Q with some leaf of F {\displaystyle {\mathcal {F}}} . Since Q

300-496: A , φ α } α ∈ A {\displaystyle {\mathcal {U}}=\left\{U_{a},\varphi _{\alpha }\right\}_{\alpha \in A}} be a regular foliated atlas of codimension q . Define an equivalence relation on M by setting x ~ y if and only if either there is a U {\displaystyle {\mathcal {U}}} -plaque P 0 such that x , y ∈ P 0 or there

400-399: A , meets U in either the empty set or a countable collection of subspaces whose images under φ {\displaystyle \varphi } in φ ( M a ∩ U ) {\displaystyle \varphi (M_{a}\cap U)} are p -dimensional affine subspaces whose first n − p coordinates are constant. Locally, every foliation

500-406: A dimension - p foliation F {\displaystyle {\mathcal {F}}} of an n -dimensional manifold M that is a covered by charts U i together with maps such that for overlapping pairs U i , U j the transition functions φ ij  : R → R defined by take the form where x denotes the first q = n − p coordinates, and y denotes

600-417: A such that all real points within distance a of x are in U (because U contains no non-rational numbers). Open sets have a fundamental importance in topology . The concept is required to define and make sense of topological space and other topological structures that deal with the notions of closeness and convergence for spaces such as metric spaces and uniform spaces . Every subset A of

700-420: A coherent refinement that is regular. Fix a metric on M and a foliated atlas W . {\displaystyle {\mathcal {W}}.} Passing to a subcover , if necessary, one can assume that W = { W j , ψ j } j = 1 l {\displaystyle {\mathcal {W}}=\left\{W_{j},\psi _{j}\right\}_{j=1}^{l}}

800-476: A collection that has the property of containing every union of its members, every finite intersection of its members, the empty set , and the whole set itself. A set in which such a collection is given is called a topological space , and the collection is called a topology . These conditions are very loose, and allow enormous flexibility in the choice of open sets. For example, every subset can be open (the discrete topology ), or no subset can be open except

900-1226: A finite subatlas U = { U i , φ i } i = 1 N {\displaystyle {\mathcal {U}}=\left\{U_{i},\varphi _{i}\right\}_{i=1}^{N}} of {( U x , φ x ) | x ∈ M }. If U i ∩ U j ≠ 0, then diam( U i ∪ U j ) < ε, and so there is an index k such that U ¯ i ∪ U ¯ j ⊆ W k . {\displaystyle {\overline {U}}_{i}\cup {\overline {U}}_{j}\subseteq W_{k}.} Distinct plaques of U ¯ i {\displaystyle {\overline {U}}_{i}} (respectively, of U ¯ j {\displaystyle {\overline {U}}_{j}} ) lie in distinct plaques of W k . Hence each plaque of U ¯ i {\displaystyle {\overline {U}}_{i}} has interior meeting at most one plaque of U ¯ j {\displaystyle {\overline {U}}_{j}} and vice versa. By construction, U {\displaystyle {\mathcal {U}}}

1000-549: A foliation. For a slightly more geometrical definition, p -dimensional foliation F {\displaystyle {\mathcal {F}}} of an n -manifold M may be thought of as simply a collection { M a } of pairwise-disjoint, connected, immersed p -dimensional submanifolds (the leaves of the foliation) of M , such that for every point x in M , there is a chart ( U , φ ) {\displaystyle (U,\varphi )} with U homeomorphic to R containing x such that every leaf, M

1100-600: A foliation. More specifically, if U {\displaystyle {\mathcal {U}}} and V {\displaystyle {\mathcal {V}}} are foliated atlases on M and if U {\displaystyle {\mathcal {U}}} is associated to a foliation F {\displaystyle {\mathcal {F}}} then U {\displaystyle {\mathcal {U}}} and V {\displaystyle {\mathcal {V}}} are coherent if and only if V {\displaystyle {\mathcal {V}}}

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1200-611: A generalization of spaces equipped with a notion of distance, which are called metric spaces . In the set of all real numbers , one has the natural Euclidean metric ; that is, a function which measures the distance between two real numbers: d ( x , y ) = | x − y | . Therefore, given a real number x , one can speak of the set of all points close to that real number; that is, within ε of x . In essence, points within ε of x approximate x to an accuracy of degree ε . Note that ε > 0 always but as ε becomes smaller and smaller, one obtains points that approximate x to

1300-511: A greater degree of accuracy. Bearing this in mind, one may define the remaining axioms that the family of sets about x is required to satisfy. Several definitions are given here, in an increasing order of technicality. Each one is a special case of the next one. A subset U {\displaystyle U} of the Euclidean n -space R is open if, for every point x in U {\displaystyle U} , there exists

1400-433: A higher and higher degree of accuracy. For example, if x = 0 and ε = 1, the points within ε of x are precisely the points of the interval (−1, 1); that is, the set of all real numbers between −1 and 1. However, with ε = 0.5, the points within ε of x are precisely the points of (−0.5, 0.5). Clearly, these points approximate x to a greater degree of accuracy than when ε = 1. The previous discussion shows, for

1500-445: A method to distinguish two points . For example, if about one of two points in a topological space , there exists an open set not containing the other (distinct) point, the two points are referred to as topologically distinguishable . In this manner, one may speak of whether two points, or more generally two subsets , of a topological space are "near" without concretely defining a distance . Therefore, topological spaces may be seen as

1600-472: A more precise definition of foliation, it is necessary to define some auxiliary elements. A rectangular neighborhood in R is an open subset of the form B = J 1 × ⋅⋅⋅ × J n , where J i is a (possibly unbounded) relatively open interval in the i th coordinate axis. If J 1 is of the form ( a ,0], it is said that B has boundary In the following definition, coordinate charts are considered that have values in R × R , allowing

1700-419: A neighborhood U and a system of local, class C coordinates x =( x , ⋅⋅⋅, x ) : U → R such that for each leaf L α , the components of U ∩ L α are described by the equations x =constant, ⋅⋅⋅, x =constant. A foliation is denoted by F {\displaystyle {\mathcal {F}}} ={ L α } α∈ A . The notion of leaves allows for an intuitive way of thinking about

1800-488: A neighborhood in which the formula is independent of x β . The main use of foliated atlases is to link their overlapping plaques to form the leaves of a foliation. For this and other purposes, the general definition of foliated atlas above is a bit clumsy. One problem is that a plaque of ( U α , φ α ) can meet multiple plaques of ( U β , φ β ). It can even happen that a plaque of one chart meets infinitely many plaques of another chart. However, no generality

1900-430: A one-to-one correspondence between the set of foliations on M and the set of coherence classes of foliated atlases or, in other words, a foliation F {\displaystyle {\mathcal {F}}} of codimension q and class C on M is a coherence class of foliated atlases of codimension q and class C on M . By Zorn's lemma , it is obvious that every coherence class of foliated atlases contains

2000-408: A part of the definition of a regular foliated atlas. This makes possible another definition of foliations in terms of regular foliated atlases. To this end, one has to prove first that every regular foliated atlas of codimension q is associated to a unique foliation F {\displaystyle {\mathcal {F}}} of codimension q . Let U = { U

2100-454: A positive real number ε (depending on x ) such that any point in R whose Euclidean distance from x is smaller than ε belongs to U {\displaystyle U} . Equivalently, a subset U {\displaystyle U} of R is open if every point in U {\displaystyle U} is the center of an open ball contained in U . {\displaystyle U.} An example of

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2200-392: A preopen set is called preclosed . The complement of a β-open set is called β-closed . The complement of a sequentially open set is called sequentially closed . A subset S ⊆ X {\displaystyle S\subseteq X} is sequentially closed in X {\displaystyle X} if and only if S {\displaystyle S}

2300-402: A real-valued smooth function ( scalar field ) whose gradient is everywhere non-zero; this smooth function is moreover usually assumed to be a time function, meaning that its gradient is everywhere time-like , so that its level-sets are all space-like hypersurfaces. In deference to standard mathematical terminology, these hypersurface are often called the leaves (or sometimes slices ) of

2400-551: A rectangular neighborhood in R and B τ {\displaystyle B_{\tau }} a rectangular neighborhood in R . The set P y = φ ( B τ × { y }), where y ∈ B ⋔ {\displaystyle y\in B_{\pitchfork }} , is called a plaque of this foliated chart. For each x ∈ B τ , the set S x = φ ({ x } × B ⋔ {\displaystyle B_{\pitchfork }} )

2500-404: A regular, foliated atlas, P 0 meets only finitely many other plaques. That is, there are only finitely many plaque chains { P 0 , P i } of length 1. By induction on the length p of plaque chains that begin at P 0 , it is similarly proven that there are only finitely many of length ≤ p. Since every U {\displaystyle {\mathcal {U}}} -plaque in L is, by

2600-399: A set X endowed with a topology τ {\displaystyle \tau } as "the topological space X " rather than "the topological space ( X , τ ) {\displaystyle (X,\tau )} ", despite the fact that all the topological data is contained in τ . {\displaystyle \tau .} If there are two topologies on

2700-779: A subatlas, it is assumed that W = { W j , ψ j } j = 0 ∞ {\displaystyle {\mathcal {W}}=\left\{W_{j},\psi _{j}\right\}_{j=0}^{\infty }} is countable and a strictly increasing sequence { n l } l = 0 ∞ {\displaystyle \left\{n_{l}\right\}_{l=0}^{\infty }} of positive integers can be found such that W l = { W j , ψ j } j = 0 n l {\displaystyle {\mathcal {W}}_{l}=\left\{W_{j},\psi _{j}\right\}_{j=0}^{n_{l}}} covers K l . Let δ l denote

2800-764: A subset S {\displaystyle S} of a topological space ( X , τ ) {\displaystyle (X,\tau )} is called clopen if both S {\displaystyle S} and its complement X ∖ S {\displaystyle X\setminus S} are open subsets of ( X , τ ) {\displaystyle (X,\tau )} ; or equivalently, if S ∈ τ {\displaystyle S\in \tau } and X ∖ S ∈ τ . {\displaystyle X\setminus S\in \tau .} In any topological space ( X , τ ) , {\displaystyle (X,\tau ),}

2900-519: A subset of R that is not open is the closed interval [0,1] , since neither 0 - ε nor 1 + ε belongs to [0,1] for any ε > 0 , no matter how small. A subset U of a metric space ( M , d ) is called open if, for any point x in U , there exists a real number ε > 0 such that any point y ∈ M {\displaystyle y\in M} satisfying d ( x , y ) < ε belongs to U . Equivalently, U

3000-767: A topological space X {\displaystyle X} is called a regular open set if Int ⁡ ( S ¯ ) = S {\displaystyle \operatorname {Int} \left({\overline {S}}\right)=S} or equivalently, if Bd ⁡ ( S ¯ ) = Bd ⁡ S {\displaystyle \operatorname {Bd} \left({\overline {S}}\right)=\operatorname {Bd} S} , where Bd ⁡ S {\displaystyle \operatorname {Bd} S} , Int ⁡ S {\displaystyle \operatorname {Int} S} , and S ¯ {\displaystyle {\overline {S}}} denote, respectively,

3100-444: A topological space X contains a (possibly empty) open set; the maximum (ordered under inclusion) such open set is called the interior of A . It can be constructed by taking the union of all the open sets contained in A . A function f : X → Y {\displaystyle f:X\to Y} between two topological spaces X {\displaystyle X} and Y {\displaystyle Y}

Belén Metamorphic Complex - Misplaced Pages Continue

3200-456: A topology on X {\displaystyle X} that is finer than τ . {\displaystyle \tau .} A topological space X {\displaystyle X} is Hausdorff if and only if every compact subspace of X {\displaystyle X} is θ-closed. A space X {\displaystyle X} is totally disconnected if and only if every regular closed subset

3300-460: A topology without any distance is given by manifolds , which are topological spaces that, near each point, resemble an open set of a Euclidean space , but on which no distance is defined in general. Less intuitive topologies are used in other branches of mathematics; for example, the Zariski topology , which is fundamental in algebraic geometry and scheme theory . Intuitively, an open set provides

3400-553: A unique maximal foliated atlas. Thus, Definition. A foliation of codimension q and class C on M is a maximal foliated C -atlas of codimension q on M . In practice, a relatively small foliated atlas is generally used to represent a foliation. Usually, it is also required this atlas to be regular. In the chart U i , the stripes x = constant match up with the stripes on other charts U j . These submanifolds piece together from chart to chart to form maximal connected injectively immersed submanifolds called

3500-445: Is continuous if the preimage of every open set in Y {\displaystyle Y} is open in X . {\displaystyle X.} The function f : X → Y {\displaystyle f:X\to Y} is called open if the image of every open set in X {\displaystyle X} is open in Y . {\displaystyle Y.} An open set on

3600-474: Is has the formula Similar assertions hold also for open charts (without the overlines). The transverse coordinate map y α can be viewed as a submersion and the formulas y α = y α ( y β ) can be viewed as diffeomorphisms These satisfy the cocycle conditions . That is, on y δ ( U α ∩ U β ∩ U δ ), and, in particular, Using the above definitions for coherence and regularity it can be proven that every foliated atlas has

3700-474: Is a stub . You can help Misplaced Pages by expanding it . Foliation In mathematics ( differential geometry ), a foliation is an equivalence relation on an n -manifold , the equivalence classes being connected, injectively immersed submanifolds , all of the same dimension p , modeled on the decomposition of the real coordinate space R into the cosets x + R of the standardly embedded subspace R . The equivalence classes are called

3800-581: Is a submersion allowing the following Definition. Let M and Q be manifolds of dimension n and q ≤ n respectively, and let f  : M → Q be a submersion, that is, suppose that the rank of the function differential (the Jacobian ) is q . It follows from the Implicit Function Theorem that ƒ induces a codimension- q foliation on M where the leaves are defined to be the components of f ( x ) for x ∈ Q . This definition describes

3900-623: Is a coherent refinement of W {\displaystyle {\mathcal {W}}} and is a regular foliated atlas. If M is not compact, local compactness and second countability allows one to choose a sequence { K i } i = 0 ∞ {\displaystyle \left\{K_{i}\right\}_{i=0}^{\infty }} of compact subsets such that K i ⊂ int K i +1 for each i ≥ 0 and M = ⋃ i = 1 ∞ K i . {\displaystyle M=\bigcup _{i=1}^{\infty }K_{i}.} Passing to

4000-642: Is a foliated C -atlas. Coherence of foliated atlases is an equivalence relation. Reflexivity and symmetry are immediate. To prove transitivity let U ≈ V {\displaystyle {\mathcal {U}}\thickapprox {\mathcal {V}}} and V ≈ W {\displaystyle {\mathcal {V}}\thickapprox {\mathcal {W}}} . Let ( U α , x α , y α ) ∈ U {\displaystyle {\mathcal {U}}} and ( W λ , x λ , y λ ) ∈ W {\displaystyle {\mathcal {W}}} and suppose that there

4100-596: Is a model of a foliated manifold with a corner separating the tangential boundary from the transverse boundary. A foliated atlas of codimension q and class C (0 ≤ r ≤ ∞) on the n -manifold M is a C -atlas U = { ( U α , φ α ) ∣ α ∈ A } {\displaystyle {\mathcal {U}}=\{(U_{\alpha },\varphi _{\alpha })\mid \alpha \in A\}} of foliated charts of codimension q which are coherently foliated in

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4200-400: Is a point w ∈ U α ∩ W λ . Choose ( V δ , x δ , y δ ) ∈ V {\displaystyle {\mathcal {V}}} such that w ∈ V δ . By the above remarks, there is a neighborhood N of w in U α ∩ V δ ∩ W λ such that and hence Since w ∈ U α ∩ W λ is arbitrary, it can be concluded that y α ( x λ , y λ )

4300-609: Is a regular closed set, where by definition a subset S {\displaystyle S} of X {\displaystyle X} is called a regular closed set if Int ⁡ S ¯ = S {\displaystyle {\overline {\operatorname {Int} S}}=S} or equivalently, if Bd ⁡ ( Int ⁡ S ) = Bd ⁡ S . {\displaystyle \operatorname {Bd} \left(\operatorname {Int} S\right)=\operatorname {Bd} S.} Every regular open set (resp. regular closed set)

4400-467: Is a semi-preopen (resp. semi-open, preopen, b-open) set. Preopen sets need not be semi-open and semi-open sets need not be preopen. Arbitrary unions of preopen (resp. α-open, b-open, semi-preopen) sets are once again preopen (resp. α-open, b-open, semi-preopen). However, finite intersections of preopen sets need not be preopen. The set of all α-open subsets of a space ( X , τ ) {\displaystyle (X,\tau )} forms

4500-441: Is a sequence L = { P 0 , P 1 ,⋅⋅⋅, P p } of U {\displaystyle {\mathcal {U}}} -plaques such that x ∈ P 0 , y ∈ P p , and P i ∩ P i -1 ≠ ∅ with 1 ≤ i ≤ p . The sequence L will be called a plaque chain of length p connecting x and y . In the case that x , y ∈ P 0 , it is said that { P 0 } is a plaque chain of length 0 connecting x and y . The fact that ~

4600-810: Is a subset of W and φ = ψ | U then, if φ ( U ) = B τ × B ⋔ , {\displaystyle \varphi (U)=B_{\tau }\times B_{\pitchfork },} it can be seen that ψ | U ¯ {\displaystyle \psi |{\overline {U}}} , written φ ¯ {\displaystyle {\overline {\varphi }}} , carries U ¯ {\displaystyle {\overline {U}}} diffeomorphically onto B ¯ τ × B ¯ ⋔ . {\displaystyle {\overline {B}}_{\tau }\times {\overline {B}}_{\pitchfork }.} A foliated atlas

4700-517: Is a topology on X {\displaystyle X} with the property that every non-empty proper subset S {\displaystyle S} of X {\displaystyle X} is either an open subset or else a closed subset, but never both; that is, if ∅ ≠ S ⊊ X {\displaystyle \varnothing \neq S\subsetneq X} (where S ≠ X {\displaystyle S\neq X} ) then exactly one of

4800-491: Is a union of plaques and the foliation by plaques is tangent to the boundary. If ∂B τ ≠ ∅ = ∂ B ⋔ {\displaystyle B_{\pitchfork }} , then ∂U = ∂ ⋔ U {\displaystyle \partial _{\pitchfork }U} is a union of transversals and the foliation is transverse to the boundary. Finally, if ∂ B ⋔ {\displaystyle B_{\pitchfork }} ≠ ∅ ≠ ∂B τ , this

4900-473: Is also associated to F {\displaystyle {\mathcal {F}}} . If V {\displaystyle {\mathcal {V}}} is also associated to F {\displaystyle {\mathcal {F}}} , every leaf L is a union of V {\displaystyle {\mathcal {V}}} -plaques and of U {\displaystyle {\mathcal {U}}} -plaques. These plaques are open subsets in

5000-438: Is an equivalence relation is clear. It is also clear that each equivalence class L is a union of plaques. Since U {\displaystyle {\mathcal {U}}} -plaques can only overlap in open subsets of each other, L is locally a topologically immersed submanifold of dimension n − q . The open subsets of the plaques P ⊂ L form the base of a locally Euclidean topology on L of dimension n − q and L

5100-410: Is an open subset (resp. is a closed subset) although in general, the converses are not true. Throughout, ( X , τ ) {\displaystyle (X,\tau )} will be a topological space. A subset A ⊆ X {\displaystyle A\subseteq X} of a topological space X {\displaystyle X} is called: The complement of

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5200-401: Is an open subset of the rational numbers , but not of the real numbers . This is because when the surrounding space is the rational numbers, for every point x in U , there exists a positive number a such that all rational points within distance a of x are also in U . On the other hand, when the surrounding space is the reals, then for every point x in U there is no positive

5300-421: Is associated to F {\displaystyle {\mathcal {F}}} and that V ≈ U {\displaystyle {\mathcal {V}}\approx {\mathcal {U}}} , let Q be a V {\displaystyle {\mathcal {V}}} -plaque. If L is a leaf of F {\displaystyle {\mathcal {F}}} and w ∈ L ∩ Q , let P ∈ L be

5400-436: Is called a transversal of the foliated chart. The set ∂ τ U = φ ( B τ × ( ∂ B ⋔ {\displaystyle B_{\pitchfork }} )) is called the tangential boundary of U and ∂ ⋔ U {\displaystyle \partial _{\pitchfork }U} = φ (( ∂B τ ) × B ⋔ {\displaystyle B_{\pitchfork }} )

5500-440: Is called a topological space . Infinite intersections of open sets need not be open. For example, the intersection of all intervals of the form ( − 1 / n , 1 / n ) , {\displaystyle \left(-1/n,1/n\right),} where n {\displaystyle n} is a positive integer, is the set { 0 } {\displaystyle \{0\}} which

5600-429: Is called a local transversal section of the foliation. Note that due to monodromy global transversal sections of the foliation might not exist. The case r = 0 is rather special. Those C foliations that arise in practice are usually "smooth-leaved". More precisely, they are of class C , in the following sense. Open set In mathematics , an open set is a generalization of an open interval in

5700-452: Is called the transverse boundary of U . The foliated chart is the basic model for all foliations, the plaques being the leaves. The notation B τ is read as " B -tangential" and B ⋔ {\displaystyle B_{\pitchfork }} as " B -transverse". There are also various possibilities. If both B ⋔ {\displaystyle B_{\pitchfork }} and B τ have empty boundary,

5800-409: Is clearly connected in this topology. It is also trivial to check that L is Hausdorff . The main problem is to show that L is second countable . Since each plaque is 2nd countable, the same will hold for L if it is shown that the set of U {\displaystyle {\mathcal {U}}} -plaques in L is at most countably infinite. Fix one such plaque P 0 . By the definition of

5900-410: Is connected, L ∩ Q = Q . Finally, Q is an arbitrary V {\displaystyle {\mathcal {V}}} -plaque, and so V {\displaystyle {\mathcal {V}}} is associated to F {\displaystyle {\mathcal {F}}} . It is now obvious that the correspondence between foliations on M and their associated foliated atlases induces

6000-518: Is equal to its sequential closure , which by definition is the set SeqCl X ⁡ S {\displaystyle \operatorname {SeqCl} _{X}S} consisting of all x ∈ X {\displaystyle x\in X} for which there exists a sequence in S {\displaystyle S} that converges to x {\displaystyle x} (in X {\displaystyle X} ). Using

6100-603: Is equivalent to requiring that, if U α ∩ U β ≠ ∅, the transverse coordinate changes y ¯ α = y ¯ α ( x ¯ β , y ¯ β ) {\displaystyle {\overline {y}}_{\alpha }={\overline {y}}_{\alpha }\left({\overline {x}}_{\beta },{\overline {y}}_{\beta }\right)} be independent of x ¯ β . {\displaystyle {\overline {x}}_{\beta }.} That

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6200-453: Is finite. Let ε > 0 be a Lebesgue number for W . {\displaystyle {\mathcal {W}}.} That is, any subset X ⊆ M of diameter < ε lies entirely in some W j . For each x ∈ M , choose j such that x ∈ W j and choose a foliated chart ( U x , φ x ) such that Suppose that U x ⊂ W k , k ≠ j , and write ψ k = ( x k , y k ) as usual, where y k  : W k → R

6300-513: Is locally independent of x λ . It is thus proven that U ≈ W {\displaystyle {\mathcal {U}}\thickapprox {\mathcal {W}}} , hence that coherence is transitive. Plaques and transversals defined above on open sets are also open. But one can speak also of closed plaques and transversals. Namely, if ( U , φ ) and ( W , ψ ) are foliated charts such that U ¯ {\displaystyle {\overline {U}}} (the closure of U )

6400-566: Is lost in assuming the situation to be much more regular as shown below. Two foliated atlases U {\displaystyle {\mathcal {U}}} and V {\displaystyle {\mathcal {V}}} on M of the same codimension and smoothness class C are coherent ( U ≈ V ) {\displaystyle \left({\mathcal {U}}\thickapprox {\mathcal {V}}\right)} if U ∪ V {\displaystyle {\mathcal {U}}\cup {\mathcal {V}}}

6500-413: Is not open in the real line. A metric space is a topological space, whose topology consists of the collection of all subsets that are unions of open balls. There are, however, topological spaces that are not metric spaces. The union of any number of open sets, or infinitely many open sets, is open. The intersection of a finite number of open sets is open. A complement of an open set (relative to

6600-546: Is open if every point in U has a neighborhood contained in U . This generalizes the Euclidean space example, since Euclidean space with the Euclidean distance is a metric space. A topology τ {\displaystyle \tau } on a set X is a set of subsets of X with the properties below. Each member of τ {\displaystyle \tau } is called an open set . X together with τ {\displaystyle \tau }

6700-456: Is open in the original topology on X , but V ∩ Y {\displaystyle V\cap Y} isn't open in the original topology on X , then V ∩ Y {\displaystyle V\cap Y} is open in the subspace topology on Y . As a concrete example of this, if U is defined as the set of rational numbers in the interval ( 0 , 1 ) , {\displaystyle (0,1),} then U

6800-490: Is open) then every subset of X {\displaystyle X} is a clopen subset. For a more advanced example reminiscent of the discrete topology, suppose that U {\displaystyle {\mathcal {U}}} is an ultrafilter on a non-empty set X . {\displaystyle X.} Then the union τ := U ∪ { ∅ } {\displaystyle \tau :={\mathcal {U}}\cup \{\varnothing \}}

6900-796: Is said to be regular if By property (1), the coordinates x α and y α extend to coordinates x ¯ α {\displaystyle {\overline {x}}_{\alpha }} and y ¯ α {\displaystyle {\overline {y}}_{\alpha }} on U ¯ α {\displaystyle {\overline {U}}_{\alpha }} and one writes φ ¯ α = ( x ¯ α , y ¯ α ) . {\displaystyle {\overline {\varphi }}_{\alpha }=\left({\overline {x}}_{\alpha },{\overline {y}}_{\alpha }\right).} Property (3)

7000-532: Is taken n −1 = 0.) This creates a regular foliated atlas U = { U i , φ i } i = 1 ∞ {\displaystyle {\mathcal {U}}=\left\{U_{i},\varphi _{i}\right\}_{i=1}^{\infty }} that refines W {\displaystyle {\mathcal {W}}} and is coherent with W . {\displaystyle {\mathcal {W}}.} . Several alternative definitions of foliation exist depending on

7100-425: Is the complement of the other. The open sets of the usual Euclidean topology of the real line R {\displaystyle \mathbb {R} } are the empty set, the open intervals and every union of open intervals. If a topological space X {\displaystyle X} is endowed with the discrete topology (so that by definition, every subset of X {\displaystyle X}

7200-970: Is the transverse coordinate map. This is a submersion having the plaques in W k as level sets. Thus, y k restricts to a submersion y k  : U x → R . This is locally constant in x j ; so choosing U x smaller, if necessary, one can assume that y k | U ¯ x {\displaystyle {\overline {U}}_{x}} has the plaques of U ¯ x {\displaystyle {\overline {U}}_{x}} as its level sets. That is, each plaque of W k meets (hence contains) at most one (compact) plaque of U ¯ x {\displaystyle {\overline {U}}_{x}} . Since 1 < k < l < ∞, one can choose U x so that, whenever U x ⊂ W k , distinct plaques of U ¯ x {\displaystyle {\overline {U}}_{x}} lie in distinct plaques of W k . Pass to

7300-421: The leaves of the foliation. If one shrinks the chart U i it can be written as U ix × U iy , where U ix ⊂ R , U iy ⊂ R , U iy is homeomorphic to the plaques, and the points of U ix parametrize the plaques in U i . If one picks y 0 in U iy , then U ix × { y 0 } is a submanifold of U i that intersects every plaque exactly once. This

7400-448: The leaves of the foliation. If the manifold and/or the submanifolds are required to have a piecewise-linear , differentiable (of class C ), or analytic structure then one defines piecewise-linear, differentiable, or analytic foliations, respectively. In the most important case of differentiable foliation of class C it is usually understood that r ≥ 1 (otherwise, C is a topological foliation). The number p (the dimension of

7500-417: The real line has the characteristic property that it is a countable union of disjoint open intervals. A set might be open, closed, both, or neither. In particular, open and closed sets are not mutually exclusive, meaning that it is in general possible for a subset of a topological space to simultaneously be both an open subset and a closed subset. Such subsets are known as clopen sets . Explicitly,

7600-410: The real line . In a metric space (a set with a distance defined between every two points), an open set is a set that, with every point P in it, contains all points of the metric space that are sufficiently near to P (that is, all points whose distance to P is less than some value depending on P ). More generally, an open set is a member of a given collection of subsets of a given set,

7700-480: The case x = 0, that one may approximate x to higher and higher degrees of accuracy by defining ε to be smaller and smaller. In particular, sets of the form (− ε , ε ) give us a lot of information about points close to x = 0. Thus, rather than speaking of a concrete Euclidean metric, one may use sets to describe points close to x . This innovative idea has far-reaching consequences; in particular, by defining different collections of sets containing 0 (distinct from

7800-748: The compact case, requiring that U ¯ x {\displaystyle {\overline {U}}_{x}} be a compact subset of W j and that φ x = ψ j | U x , some j ≤ n l . Also, require that diam U ¯ x {\displaystyle {\overline {U}}_{x}} < ε l /2. As before, pass to a finite subcover { U i , φ i } i = n l − 1 + 1 n l {\displaystyle \left\{U_{i},\varphi _{i}\right\}_{i=n_{l-1}+1}^{n_{l}}} of K l ╲ {\displaystyle \diagdown } int K l -1 . (Here, it

7900-493: The coordinates formula can be changed as The condition that ( U α , x α , y α ) and ( U β , x β , y β ) be coherently foliated means that, if P ⊂ U α is a plaque, the connected components of P ∩ U β lie in (possibly distinct) plaques of U β . Equivalently, since the plaques of U α and U β are level sets of the transverse coordinates y α and y β , respectively, each point z ∈ U α ∩ U β has

8000-453: The definition of ~, reached by a finite plaque chain starting at P 0 , the assertion follows. As shown in the proof, the leaves of the foliation are equivalence classes of plaque chains of length ≤ p which are also topologically immersed Hausdorff p -dimensional submanifolds . Next, it is shown that the equivalence relation of plaques on a leaf is expressed in equivalence of coherent foliated atlases in respect to their association with

8100-926: The distance from K l to ∂ K l +1 and choose ε l > 0 so small that ε l < min{δ l /2,ε l -1 } for l ≥ 1, ε 0 < δ 0 /2, and ε l is a Lebesgue number for W l {\displaystyle {\mathcal {W}}_{l}} (as an open cover of K l ) and for W l + 1 {\displaystyle {\mathcal {W}}_{l+1}} (as an open cover of K l +1 ). More precisely, if X ⊂ M meets K l (respectively, K l +1 ) and diam X < ε l , then X lies in some element of W l {\displaystyle {\mathcal {W}}_{l}} (respectively, W l + 1 {\displaystyle {\mathcal {W}}_{l+1}} ). For each x ∈ K l ╲ {\displaystyle \diagdown } int K l -1 , construct ( U x , φ x ) as for

8200-537: The empty set ∅ {\displaystyle \varnothing } and the set X {\displaystyle X} itself are always clopen. These two sets are the most well-known examples of clopen subsets and they show that clopen subsets exist in every topological space. To see, it suffices to remark that, by definition of a topology, X {\displaystyle X} and ∅ {\displaystyle \varnothing } are both open, and that they are also closed, since each

8300-425: The fact that whenever two subsets A , B ⊆ X {\displaystyle A,B\subseteq X} satisfy A ⊆ B , {\displaystyle A\subseteq B,} the following may be deduced: Moreover, a subset is a regular open set if and only if it is preopen and semi-closed. The intersection of an α-open set and a semi-preopen (resp. semi-open, preopen, b-open) set

8400-415: The foliated chart models codimension- q foliations of n -manifolds without boundary. If one, but not both of these rectangular neighborhoods has boundary, the foliated chart models the various possibilities for foliations of n -manifolds with boundary and without corners. Specifically, if ∂ B ⋔ {\displaystyle B_{\pitchfork }} ≠ ∅ = ∂B τ , then ∂U = ∂ τ U

8500-431: The foliation. Note that while this situation does constitute a codimension-1 foliation in the standard mathematical sense, examples of this type are actually globally trivial; while the leaves of a (mathematical) codimension-1 foliation are always locally the level sets of a function, they generally cannot be expressed this way globally, as a leaf may pass through a local-trivializing chart infinitely many times, and

8600-559: The following two statements is true: either (1) S ∈ τ {\displaystyle S\in \tau } or else, (2) X ∖ S ∈ τ . {\displaystyle X\setminus S\in \tau .} Said differently, every subset is open or closed but the only subsets that are both (i.e. that are clopen) are ∅ {\displaystyle \varnothing } and X . {\displaystyle X.} A subset S {\displaystyle S} of

8700-435: The holonomy around a leaf may also obstruct the existence of a globally-consistent defining functions for the leaves. For example, while the 3-sphere has a famous codimension-1 foliation discovered by Reeb, a codimension-1 foliation of a closed manifold cannot be given by the level sets of a smooth function, since a smooth function on a closed manifold necessarily has critical points at its maxima and minima. In order to give

8800-1056: The last p co-ordinates. That is, The splitting of the transition functions φ ij into φ i j 1 ( x ) {\displaystyle \varphi _{ij}^{1}(x)} and φ i j 2 ( x , y ) {\displaystyle \varphi _{ij}^{2}(x,y)} as a part of the submersion is completely analogous to the splitting of g ¯ α β {\displaystyle {\overline {g}}_{\alpha \beta }} into y ¯ α ( y ¯ β ) {\displaystyle {\overline {y}}_{\alpha }\left({\overline {y}}_{\beta }\right)} and x ¯ α ( x ¯ β , y ¯ β ) {\displaystyle {\overline {x}}_{\alpha }\left({\overline {x}}_{\beta },{\overline {y}}_{\beta }\right)} as

8900-404: The leaves) is called the dimension of the foliation and q = n − p is called its codimension . In some papers on general relativity by mathematical physicists, the term foliation (or slicing ) is used to describe a situation where the relevant Lorentz manifold (a ( p +1)-dimensional spacetime ) has been decomposed into hypersurfaces of dimension p , specified as the level sets of

9000-438: The manifold topology of L , hence intersect in open subsets of each other. Since plaques are connected, a U {\displaystyle {\mathcal {U}}} -plaque cannot intersect a V {\displaystyle {\mathcal {V}}} -plaque unless they lie in a common leaf; so the foliated atlases are coherent. Conversely, if we only know that U {\displaystyle {\mathcal {U}}}

9100-401: The measure as being a binary condition: all things in R are equally close to 0, while any item that is not in R is not close to 0. In general, one refers to the family of sets containing 0, used to approximate 0, as a neighborhood basis ; a member of this neighborhood basis is referred to as an open set. In fact, one may generalize these notions to an arbitrary set ( X ); rather than just

9200-480: The possibility of manifolds with boundary and ( convex ) corners. A foliated chart on the n -manifold M of codimension q is a pair ( U , φ ), where U ⊆ M is open and φ : U → B τ × B ⋔ {\displaystyle \varphi :U\to B_{\tau }\times B_{\pitchfork }} is a diffeomorphism , B ⋔ {\displaystyle B_{\pitchfork }} being

9300-527: The real numbers. In this case, given a point ( x ) of that set, one may define a collection of sets "around" (that is, containing) x , used to approximate x . Of course, this collection would have to satisfy certain properties (known as axioms ) for otherwise we may not have a well-defined method to measure distance. For example, every point in X should approximate x to some degree of accuracy. Thus X should be in this family. Once we begin to define "smaller" sets containing x , we tend to approximate x to

9400-465: The same set, a set U that is open in the first topology might fail to be open in the second topology. For example, if X is any topological space and Y is any subset of X , the set Y can be given its own topology (called the 'subspace topology') defined by "a set U is open in the subspace topology on Y if and only if U is the intersection of Y with an open set from the original topology on X ." This potentially introduces new open sets: if V

9500-422: The sense that, whenever P and Q are plaques in distinct charts of U {\displaystyle {\mathcal {U}}} , then P ∩ Q is open both in P and Q . A useful way to reformulate the notion of coherently foliated charts is to write for w ∈ U α ∩ U β The notation ( U α , φ α ) is often written ( U α , x α , y α ), with On φ β ( U α ∩ U β )

9600-448: The sets (− ε , ε )), one may find different results regarding the distance between 0 and other real numbers. For example, if we were to define R as the only such set for "measuring distance", all points are close to 0 since there is only one possible degree of accuracy one may achieve in approximating 0: being a member of R . Thus, we find that in some sense, every real number is distance 0 away from 0. It may help in this case to think of

9700-427: The space itself and the empty set (the indiscrete topology ). In practice, however, open sets are usually chosen to provide a notion of nearness that is similar to that of metric spaces, without having a notion of distance defined. In particular, a topology allows defining properties such as continuity , connectedness , and compactness , which were originally defined by means of a distance. The most common case of

9800-476: The space that the topology is defined on) is called a closed set . A set may be both open and closed (a clopen set ). The empty set and the full space are examples of sets that are both open and closed. A set can never been considered as open by itself. This notion is relative to a containing set and a specific topology on it. Whether a set is open depends on the topology under consideration. Having opted for greater brevity over greater clarity , we refer to

9900-428: The topological boundary , interior , and closure of S {\displaystyle S} in X {\displaystyle X} . A topological space for which there exists a base consisting of regular open sets is called a semiregular space . A subset of X {\displaystyle X} is a regular open set if and only if its complement in X {\displaystyle X}

10000-422: The way through which the foliation is achieved. The most common way to achieve a foliation is through decomposition reaching to the following Definition. A p -dimensional, class C foliation of an n -dimensional manifold M is a decomposition of M into a union of disjoint connected submanifolds { L α } α∈ A , called the leaves of the foliation, with the following property: Every point in M has

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