§ 5 Compact point sets and junction point sets

A     compact point set

[ Compact point set and its properties ] Assuming that S is a point set in a topological space, any closed and open set family of S must have a finite closed cap family, then S is called a compact point set, or S Tight .

A compact point set has the properties:

The image of a 1°      compact point set under continuous transformation is compact .

2° The necessary and sufficient conditions for a point set       S in a topological space to be compact are: Any infinite subset Q of S has at least one foci x 0 S , and for any neighborhood G of x 0 ,

card ( GQ )= card ( Q )

( See § 2 for the definition of card ).

3° The necessary and sufficient conditions for a point set       S in a topological space to be compact are: Every point network in S has a subnet that converges to a point x 0 S .

Relative closed set compaction of 4°      compact point sets .

A compact point set in T 2 space is closed .

Packet compaction of compact point sets in a 6°      regular space .

The necessary and sufficient conditions for a point set S in the scale space to be compact are: S is fully bounded (that is, for any positive number r , S can always be covered by a finite number of spheres of radius equal to r ) and complete . In particular, n The necessary and sufficient conditions for the compactness of the point set S in the dimensional Euclidean space En are : S is bounded and closed .

The necessary and sufficient conditions for a scatter set S to be compact are: S is finite .

9° The necessary and sufficient conditions for a point set       S in a topological space to be compact are: The general set of any relatively closed set of S that is finitely intersected (that is, the general set of any finite set in the family is not empty) is not empty .

10 o  Cantor's theorem assumes that < B n | n ω > is a non-empty relatively closed set of compact point sets S , each B n é B n +1 , ( n = 0 , 1 , × × × ) then .

11 o  The topological product compactness of a family of Tikhonov's theorems for compact spaces .

[ Compact - open topology of transformation families ] Assume that X and Y are two topological spaces . In the overlapping set X Y (the set of all transformations that transform X into Y ), construct a topology as follows: For any open set V in Y , any compact set K in X , all transformations W ( K , V ) that transform K into V define as an open set in X Y , and from all such open sets W ( K , V ) multiply a topology . This topology is called the compact - open topology of X Y.

Assuming that X is a topological space and Y is a scale space, then for the totality C of all continuous transformations of X and Y , compared with other topologies of X and Y , the characteristics of compact - open topology are: any one A necessary and sufficient condition for the convergence of the continuous transformation network < f p | p Q > is: for any compact set in X, < f p ( x ) | p Q > uniformly converges in K.

Therefore, the whole C of all continuous transformations is a closed set under the compact - open topology of X Y.

" Y is the scale space" in 1 ° can be changed to " Y is the uniform space" .

Askori's theorem assumes that X is a regular local compact space * , Y is a scale space with j as the scale, and C represents the totality of all continuous transformations that transform X into Y. Then a subfamily F of C The necessary and sufficient conditions for compactness under the compact - open topology of X Y are :

(i)  F is a relatively closed set of C ;

(ii) For each point x in X , the bag of the entirety of the images of x under all transformations belonging to F is a compact set in Y ;

(iii) F is equally continuous (that is, assuming that x 0 is a point in X , if for any positive number e there is always a neighborhood V of x 0 such that for any x V and any f F , there is j ( f ( x ), f ( x 0 )) < e ) .

" Y is a scale space" in the theorem can be changed to " Y is a T2 consistent space" .

[ Compactization ] Assume that X is a topological space and X* is a compact space . If there is a homeomorphic transformation f that transforms X into X* , and f ( X ) is dense everywhere in X* , then X* is called (or < f , X* > ) is a compactification of X. At this time, X and f ( X ) are often confused, so X is regarded as a subset of X* .

Single-point compactification  Assume that X is a topological space, and the set of bearing points of X is denoted as D. Anything that does not belong to D is denoted as ∞ . Use the entirety of the following two subsets of D{ } as the topology Subunit: ( i ) an open set in X , ( ii ) the complement of any compact closed set in X in D {} . From this subunit we get a topology τ * of D{ } . Topological space X* = < D{ }, τ* > is a compact space . Under the identity transformation, X becomes homeomorphically into X* and X is dense everywhere in X* , so X* is a compactification of X, called a single-point compaction of X , ∞ is called the infinity point in X* .

The one -point compactification of a one-dimensional complex space C1 is called a complex sphere .

The one-point compactification of the one-dimensional real space R 1 is homeomorphic to the circle .

2 ° wide one - dimensional real  space _ _

- ∞, treat all of the following sets of points as topological subunits in R 1{ } {- } : ( i ) open sets in R 1 , ( ii ) ( a , )∪ { } , ( iii ) ( - , b ) ∪ {- }. Using the topology propagated by this topological subunit as topology, R 1{ } {- } is a compact space, called a wide one-dimensional space real number space, is in the identity transformation f ({ x = fA compactification of R 1 under ( x )| x R 1 }) .

In the wide one-dimensional real number space, ∞ is the aggregation point of any unbounded real number set . If f is a transformation that transforms an unbounded real number set S into a topological space, then since ∞ is a collection point of S The meaning of point, is included in the definition of limit in § 3,4 .

The set of        connection points

[ Joint point set × region × continuous domain ] Assuming that the point set in a topological space is not the sum set of two non-empty relatively closed sets without common points, then it is called a connection point set . The connected open set is called Region . A closed set of connections that contains more than one point is called a continuous domain .

Note that in § 2 , the "continuous field hypothesis" in Liuli refers to a continuous field in the one-dimensional real space R 1 .

In the above definition of join point set, "relatively closed set" can obviously also be changed to "relatively open set" .

The definition of a join point set can also be changed to "a point set that has no non-empty relative proper subsets that are both open and closed" .

[ Properties of Junction Point Sets ]

The image connection of a 1°      connection point set under continuous transformation .

If S is a set of junction points, A , then SA junction . In particular, the package junction of the set of junction points S.

If the general set of a family of connection point sets is not empty, then the sum of this family of connection point sets is connected .

There are only the following nine types of connection point sets in the one-dimensional real number space R 1 : R 1 itself, ( a , b ), ( a , b ] , [ a , b ), [ a , b ] , ( a , ∞ ), [ a , ),(- , b ),(- , b , where a and b represent any real numbers .

The property 1 ° can be regarded as an extension of the following theorem in differential calculus: Assuming that f is a continuous real function in an interval, if two values ​​a and b are obtained, then any value between a and b must be obtained .

[ A set of points connected by lines ] A closed interval of real numbers [ a , b ] has an image under a continuous transformation into a topological space X called a curve in this space X. If a point set in a topological space is If any two points of , belong to a sub-curve of this set of points, then this set of points is said to be connected by a line .

Point sets connected by lines .

[ Local connection and local line connection ] A point set in topological space, if any relative neighbor of any point of it must cover up a connected relative neighbor of this point, then this point set is called local connection . If " "connect" is replaced by "connect with lines", then this point set is said to be locally connected with lines .

For example , any open set in the n -dimensional Euclidean space En is locally connected by a line, because any open ball is connected by a line .

A necessary and sufficient condition for a locally connected point set to be connected by a line is that it is connected by a line . A sufficient and necessary condition for an open set in particular to be a region is that it is connected by a line .

[ Independent slice and no connection at all ] Assuming that a subset of a point set in a topological space is connected and is not a proper subset of other connected subsets, then this subset is called an independent slice of this point set .

A point set of a topological space is the sum set of all its independent slices, each independent slice is relatively closed, and any two different independent slices have no common point . In particular, each independent slice of an open set is a region, So it is a relatively both open and closed subsets . An open set is the sum set of a family of regions with no common point . Each independent slice of a closed set that contains more than one point is a continuous field, and a closed set is a family of two without a common point. The sum set of connected closed sets of .

Note that the sum set of infinite connected closed sets with no common point is not necessarily a closed set .

Assuming that each independent slice of a point set in a topological space contains only one point, then the point set is said to be disjoint at all . For example , all rational numbers and all irrational numbers in R 1 are disjoint at all .

§ 6 Manifolds _

[ n -dimensional real manifold ] Assuming that M is a T 2 connection space, M has a closed-capped open set family S , for each open set V S , there is a topological transformation f V to change V to an n -dimensional interval, then it is called { f V | V S } is an n -dimensional real manifold structure of M, and M is said to be an n -dimensional real manifold .

[ Local coordinate method ] Assuming that { f V | V S } is a manifold structure of manifold M , then each V S is called a coordinate region . Each f V is a local coordinate method in V, and for each point x V ,

Called the coordinate of x , the real number x k ( k =1, × × × , n ) is called the kth coordinate of x .

[ Connection relationship ] Assuming V S , V' S , VV' 1 φ , then each point x VV' has a coordinate sum under the two local coordinate methods of f V and f V ' , and their relationship can be expressed as Expressed as

( 1 )

From the definition of manifold, it is the topological transformation of changing f V ( VV' ) to f V' ( VV' ), which is called the connection relationship from the local coordinate method f V to the local coordinate method f V' .

[ Differential Structure and Differential Manifold ] Assume that { f V | V S } is a manifold structure of manifold M , which is the connection relationship from f V to f V' . The expression ( 1 ) can be rewritten as a system of equations

( 2 )

If each function in ( 2 ) exists in f V ( VV' ) with respect to the real variables x 1 , × × × , x n in the order of 1 to m partial derivatives exist and are continuous, then it is called m -order continuously differentiable . If each partial derivative of each order exists (and therefore is continuous) in f V ( V ∩ V' ), then it is said to be continuously differentiable of order. If each is in f V ( VV' ) parsed in (i.e. at every point < x 1 0 , ×In a neighborhood of × × , x n 0 >( f V ( V V' )) ,( x 1 , × × × , x n )can be expanded into a power series of n real variables), then it is calledis analytic or ω -order continuously differentiable.

A manifold structure { f V | V S } is called an m -order differential structure if all the articulation relations of it are m -order continuously differentiable (and therefore are invertible m -order continuously differentiable) . If a manifold All connections of a structure are continuously differentiable of order ∞, then it is called a differential structure of order ∞ . If all connections of a manifold structure are analytic, then it is called an analytic structure .

Assuming that a manifold structure { f V | V S } of a real manifold M is an m -order differential structure or an infinity-order differential structure or a real analytical structure, then M is an m -order differential manifold or an infinity-order under this structure, respectively. Differential manifolds or real analytical manifolds .

[ The equivalence of differential structure ] Suppose { f V | V S } is an m -order differential structure of the manifold M. Also suppose that G is an open set in M , and f is a function defined in G. For each point x GV ( V S ), f ( x ) can be expressed as

f ( x ) = ( )

Assuming that the partial derivatives of each order from 1 to k ( 0 km ) are continuous with respect to these n real variables , then f is said to be k -order continuous differentiable in GV. If f is in every GV ( V S ) can be continuously differentiated at level k , then f can be continuously differentiated at level k in G , denoted as f C k ( G ), since all the connection relationships of { f V | V S } can be continuously differentiated at level m , and suppose 0 km , the above definition does not contradict any x V V ' ( V S , V ' S ) .

Let { f V | V S } and { f W | W π } be two m -order differential structures of the manifold M (where π is also a closed-open family of M ) . If { f V | V S } { f W | W π } is an m -order differential structure of M , then { f V | V S } and { f W | W π } is equivalent . The necessary and sufficient conditions for the equivalence of two m -order differential structures of manifold M are: for any open set G in M , each function family C k ( G ) ( k =0 , 1 , × × × , m ) are consistent .

The concept of the equivalence of two ∞ order differential structures or the equivalence of two real analytic structures of a manifold M can also be similarly defined .

[ Orientable Manifold ] Suppose that a neighborhood of a point in n -dimensional real space is transformed into a neighborhood of a point by a topological transformation f , that is, f ( ) = . If f is reversibly differentiated continuously, and the Jacobian formula

Then say f to hold the stance at this point .

If f is not differentiable, substituting the difference quotient (see Chapter 5) for the partial derivative also allows f to hold the pose at one point .

Assuming that the manifold M has a manifold structure, any of its connection relations maintains a posture at each point in the respective defining open set, then the manifold structure is called a directional manifold structure, and the manifold M is defined by it. towards .

Assuming { f V | V S } and { f W | W π } are two directed structures of manifold M , and { f V | V S } { f W | W π } is also an directed structure of M , then They are said to be oriented in the same direction .

Suppose { f V | V S } is a directed structure of the manifold M , and suppose

f V ( x ) = x    V

then

g V ( x ) = x   V

is another directional structure { g V | V S } . Obviously { f V | V S } { g V | V S } is no longer a directional structure . Then { f V | V S } and { g V | V S } is in the opposite direction .

Therefore, if the manifold M has a directional structure, then M has two types of directional structures, the structures of the same type have the same orientation, and the orientations of different types are opposite . Therefore, when the type of directional structure is not specified, it is only said that there are The manifold M of a directional structure is orientable .

It can be proved that any manifold structure of an orientable manifold can become a directional structure as long as a part of the local coordinate method is modified as in the above specification g V , and the orientation can be consistent with this type of directional structure, or it can be consistent with another directional structure. class consistency .

Therefore, although the differential structures of an orientable differential manifold are not all oriented, each equivalence class of differential structures must contain two directional structures, and the orientations are opposite to each other .

The simplest example of an unorientable manifold is the " Möbius strip", which is a one-sided surface whose model can be obtained by twisting 180 ° , aligning the sides ad and cb is glued together, a and c are superimposed, b and d are superimposed .

[ Complex Analytical Manifold ] Rewrite the point < x 1 , x 2 > in the two-dimensional real space R 2 as a complex number x 1 + ix 2 , then we get the one-dimensional complex space C 1 , and C 1 is the general topology of R 2 When topology . The general topology of R 2 can use the whole of two-dimensional intervals as the basis, or the whole of open circles as the basis . In C 1 , for the convenience of notation, it is more common to use the latter as the basis . A complex number z 0 as the center An open circle can be expressed as { z | z C 1 and | z - - z 0 |< r } , where the radius r is a positive number . The topological product of n C 1s is called an n - dimensional complex space C n , and a basis of the topology of C n is the direct result of the open circles in n C 1s The totality of products, the direct product of n open circles is called an n -fold column .

In the definition of manifold, if " n -dimension interval" is changed to " n -column", it becomes the definition of "complex manifold" .

The complex analytic structure of a complex manifold is defined in the same way as the real analytic structure of a real manifold . In particular, a one-dimensional complex analytic manifold is called a Riemann surface, which is an important concept in the theory of complex variables (see Chapter 10) .

[ The Theorem of Existence ]

Theorem 1  Some manifolds cannot have a first-order differential structure .

Note that when m m' 1 , by definition, the m -order differential structure must be the m' -order differential structure, so the manifold mentioned in Theorem 1 must not have any order of differential structure .

Theorem 2 An m ( m 1 )-order differential structure of the second countable real manifold must have an equivalent ∞-order differential structure (equivalent here means to treat it as an m -order differential structure) .

Theorem 3 The sphere in 8    -dimensional Euclidean space has an unequal differential structure .

It is known from Theorem 3 that manifolds of unequal differential structures do exist . As for the knowledge of the differential structure problem of the sphere itself, it has now been proved that the number of different equivalence classes of each differential structure dn is equal to some The number of elements of a finite group, and there are many d n have been calculated, such as

 n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 d n 1 1 ? 1 1 1 28 2 8 6 992 1 3 2 16256

It can be seen from the table that there are a total of 28 types of differential structures, and the different types are not equivalent . d 3 , that is , the number of differential structures that are not equivalent, has not been calculated .

Theorem 4 Any m ( 1 m ∞ ) order differential structure of the second countable and orientable two-dimensional real manifold is equivalent to a complex analytic structure (the latter is regarded as an m order real differential structure) .

* Every point in a topological space has a compact neighborhood, which is called a locally compact space.