**§ ****3 ****Topological spaces**** ****
**

1.
Basic Concepts

[ Topology and Topological Spaces ] Suppose *D* is a set, *τ* (that is , every element of *τ is a subset of **D* ), and satisfies the condition:
_{}_{}

(i) *φ **τ* , *D **τ* ;_{}_{}

(ii) the sum of any family of sets belonging to *τ belongs to **τ* ;

(iii) the general set of any finite set of sets belonging to *τ belongs to **τ* ,

Then *τ* is called a topology of *D* , and the ordered pair < *D* , *τ* > (see § 1 , 2) is called a topological space .

Assuming that *X* =< *D* , *τ* > is a topological space, then each element of *D* is called a point in *X* , and each subset of *D is called a point set in **X* , in particular, *D* is called a bearing point of *X* set . Each element of *τ* (which is a special subset of *D* ) is called an open set in *X , and **τ* is called the topology of *X.*

In the case of not causing misunderstanding, a topological space is often confused with its set of bearing points .

[ Solid topology and scattered topology ] Note that the topology of any set *D* always exists . For example , { *φ* , *D* } is a topology of D, called *the* solidified topology of *D* , and < *D* , { *φ* , *D* } > is called A solidified space, in this solidified space, the open set is only *φ* and *D.* There is also a topology of D, called *the* scattered topology of D, *<* D *,* > is called the scattered space, in this scattered space, any point set All are open episodes . _{}* *_{}

[ Induced topology and topological subspace ] Assume that *X* is a topological space, and *A* is a point set in X. *If* any open set in *X* and the general set of A are called *a* relative open set of *A* , then all the sets of *A* are called relative open sets of A. The relatively open set *τ* ' is a topology of *A* , called the induced topology of *A* , and < *A* , *τ* '> is called a topological subspace of *X.*

Note that when we say topological subspace, its topology must refer to induced topology .

[ Topological thickness ] Assuming that both *τ *_{1} and *τ *_{2} are topologies of set *D , **τ *_{1 }ì *τ *_{2} , then *τ *_{1} is thicker than *τ *_{2} , or *τ *_{2} is thinner than *τ *_{1} . _{}_{}_{}_{}_{}_{}_{}_{}

Each topology of a set *D* is a subset of, and therefore is an element of, and applying the axiom of partition, the totality of all topologies of a set *D is a set, called the topological family of **D* , and the thickness relation is this topological family There is a small-big relationship in D, but it is not an order, because the different topologies of *D* may not necessarily be thicker . Therefore, the topological family of *D* is a branch set according to this thickness relationship . However, when *D* has more than one element, *D* There must be the thickest topology, which is the solidification topology, and there must be the thinnest topology, which is the scattered topology ._{}_{}

[ Determination of Topological Subunits and Topology ] Although the whole of any family of subsets of a set *D satisfies the conditions defined above*

* For real numbers, see Chapter 1, § 1 , 1 .

The components ( i ), ( ii ) and ( iii ) can be taken as topology, but it is often inconvenient to verify whether these conditions are satisfied . Usually, the concept of topological subunits is used to determine a topology .

Assuming that *s* is a family of subsets of set *D (that is, **s* ), and denote the general set of all topologies *τ* (that is, *τ *ê *s** ) of D* that cover *s** as τ *_{0} , then it is not difficult to see that *τ *_{0} is a topology and is The thickest topology that masks *s* . *τ *_{0} is called the topology that *s* reproduces, and *s* is called a subunit of *τ *_{0} .*
*_{}_{}*
*_{}_{}* *_{}* *_{}

Any topology *τ* is a topology that it reproduces, and is therefore a subunit of itself .

From this definition, we know that any family of subsets of set *D* can reproduce a unique topology .

Example 1
(One-dimensional real number space *R *^{1} ) Denote the whole of real numbers as *R *^{1} . The topology *τ *^{1} propagated by the whole of all intervals ( *a* , *b* ) (when *a *3 *b* , ( *a* , *b* ) represents an empty set) is called *R *^{1} is the general topology of R 1. Unless otherwise stated, *R *^{1} is regarded as a topological space with this general topology, called a one-dimensional real number space .^{}^{}^{}

* R *^{1} is regarded as a set and there are other topologies, in addition to solidification topology and scattered topology, such as all semi-closed intervals ( *a* , *b* ] (that is, { *x* | _{}} , when *a *3 *b* , ( *a* , *b* ] Represents an empty set) and also reproduces a topology . But these are not ordinary topologies. If you want to use these topologies, you must declare them separately .* *

[ Topological basis ] Suppose *s* is the whole of a family of open sets in a topological space *X.* If any open set in X is the sum set of a family of open sets belonging to *s **,* then *s* is called a topology of X. *It* is obvious that The topology of *X* is itself a basis of itself .
* ** *

From this definition, if *s* is a basis of the topology of the topological space *X* , then it must be a subunit of the topology of *X.*

Theorem A sufficient and necessary condition for the whole *s* of a family of subsets of a set *D to be a basis of the topology it reproduces is: for any **A **s* , any *B **s* and any *x **A* ∩ *B* , there exists *C **s** such* that *x **C **A* ∩ *B.* _ * *_{}_{}* *_{}_{}_{}_{}

Thus it can be seen that the totality of all real intervals ( *a* , *b* ) is ^{a basis of the ordinary topology of }*R1 , since any real number **x* belonging to the general set of any two intervals must belong to a subinterval of this general set, Therefore, we also know that any open set in *R1 is *^{a} sum set of intervals .^{}^{}

[ Open Neighborhood, Neighborhood and Basic Neighborhood ] Suppose a point *x* in a topological space belongs to an open set, then this open set is called an open neighborhood of *x* . Assuming that a point set covers a point neighborhood of *x* , Then call this set of points a neighborhood of *x* . Assuming that an open neighborhood of x belongs to the basis of the topology of this space, then call this open neighborhood *a* basic neighborhood of *x* .

*A point set S* in a topological space is a necessary and sufficient condition for an open set: every point belonging to *S* has at least one basic neighborhood covered by *S.*

Open sets can also be defined using the concept of basic neighborhoods . This is another way of determining topology in general using topological bases . For example, this is specified: Suppose *S* is a set of real numbers . If for any *x **S* , there is an interval ( *a* , *b )* ) let *x* ( *a* , *b* ) *S* , then *S* is called an open set . The whole of all such open sets is exactly the ordinary topology ^{of }*R1* ._{}_{}_{}^{}

[ Topological product space ] Assuming { *X ** _{h}* |

Note that { _{}| *A _{h}* is any element of

In the case of not causing misunderstanding, this topological product space is often recorded as its bearing point set , because when we speak of a topological product space, it means that its topology is a product topology ._{}

[ *n* -dimensional real number space and *n* -dimensional interval ] Denote all real numbers as *R *^{1} . From example 1 , we can see that *R *^{1} is a one-dimensional real number space . When *n* is a positive integer, the topological product space of *n **R *^{1} s is written as *R ** ^{n}* , called the

If the direct product of n intervals is *called* an *n* -dimensional interval (if one of them ( *a ** _{i}* ,

2.
Basic Topological Concepts of Point Sets

[ Interior · Exterior · Boundary · Package ] Assuming that *S* is a point set in the topological space *X* =< *D* , *τ* > , that is, *S **D , then the points in **X* can be divided into three categories relative to *S* : _{}* *

1° interior point and interior . If there exists an open set *V for a point **x* such that *x **V **S* , then *x* is called the interior point of *S.*_{}_{}

*
*The totality of all interior points of S, called *the interior of **S* , denoted as *N* ( *S* ), the interior of *S* is a subset of *S.*

2° outside point and outside . The interior point of the complement *D* \ *S of **S* is called the outside point of *S.*

*
*The totality of all outer points of S is called *the* outer of *S* , and the outer of *S* is a subset of the complement of *S.*

3 ^{o} Boundary points and boundaries . A point that is neither an interior point of S nor *an* exterior point of S is called *a* boundary point of *S.*

*
*The totality of the boundary points of S is called *the boundary of **S* , denoted as *B* ( *S* ) .

*
S* ∪ *B* ( *S* ) is called a bag of *S* , denoted as = *S* ∪ *B* ( *S* ) ._{}

The basic relationship between them is as follows:

The boundary of the point set *S* is also the boundary of the complement of *S.*

The complement of the bag of the point set *S* is the exterior of *S ; the **complement* of the bag of the complement of *S* is the interior of S.

The bag of point set *S* is the sum set of the interior of *S* and the boundary of S, that is, *=* S *∪* B *(* S *)* = N *(* S *)* ∪ *B* ( *S* ); note that the general sum is not necessarily equal, that is, = *N* ( *S* ) ∪ *B* ( *N* ( *S* ) ) does not necessarily hold ._{}_{}_{}_{}

[ Everywhere Dense and Nowhere Dense ] Assuming that *P* and *Q* are point sets in a topological space, , then *P* is said to be dense everywhere in *Q.* Assuming that the exterior of P is dense everywhere in Q *, **then* P is said to *be* nowhere in *Q* is dense . Note that " exterior of *P* " cannot be replaced by "complement of *P* " here . _{}

For example, all rational numbers are dense everywhere in the one-dimensional real number space *R *^{1.} All irrational numbers are dense everywhere in *R* 1. All integers are dense nowhere in R *1. *^{A }^{non} -empty interval ( *a* , *b* ) is both dense in *R *^{1} Nowhere is dense and nowhere is dense .^{}^{}^{}

[ Open and Closed Sets ] The concept of an open set in a topological space < *D* , *τ* > is fundamental (this section, 1), and the complement of an open set is called a closed set .

The necessary and sufficient conditions for the 1° point set *S* to be open are: *S* is equal to its interior, or that every boundary point of S does not belong *to **S* .

The necessary and sufficient conditions for the 2° point set *S* to be closed are: *S* is equal to its bag, or that every boundary point of S belongs *to **S* .

The necessary and sufficient conditions for the 3° point set *S* to be both open and closed are: The boundary of *S* is an empty set . For example, *φ* and *D* are both open and closed .

The necessary and sufficient conditions for the 4° point set *S* to be neither open nor closed are *: **B* ( *S* ) ∩ S1φ and *B* ( *S* ) *∩S1B **( **S **)* .

For example, in *R *^{1} , the non-empty interval ( *a* , *b* ) is open but not closed, the semi-closed interval ( *a* , *b* ) is not open and not closed, the closed interval [ *a* , *b* ] is closed but not open, and all rational numbers are not open and not open. closed, irrational numbers are not open and closed, integers are closed but not open, and *R *^{1} is both open and closed .

In addition, from the definition of closed sets, three properties that are relative to open sets are obtained:

1° *φ* is a closed set, *D* is a closed set;

2° The general set of any family of closed sets is a closed set;

3° The sum set of any finite number of closed sets is closed .

[ Isolate point, cluster point and derivative set ] Assuming that *S* is a point set in topological space, a point *x **S* and *x* has a neighborhood *G* such that *G* ∩ *S* = { *x* } , then *x* is called an isolated point of *S.* _{}

Suppose *y* ( representing the bag of *S* ), but *y* is not an isolated point of S, then call *y **a* cluster of *S.*_{}_{}_{}

A necessary and sufficient condition for a point *y* to be a convergent point of a point set *S* is: for any neighborhood *L of **y* , ( *L* \ { *y* } )∩ *S *1 *φ* .

It is known from the definition that any isolated point of a point set *S must be the boundary point of **S* , and any interior point of a point set must be the converging point of *S* , but it is obviously not possible to reverse it .

* The totality of the foci of S* is also called the derivative set of S, denoted as *S **'* . The bag of *S* can be expressed as:_{}

*
_{}*=

[ Isolated point set, self-dense and complete set ] For any point set *S* in a topological space, all points in this space can be divided into three categories: the outer points of *S* , the isolated points of *S* , and the aggregate points of *S* Points . Aggregate points include interior points of *S* and non-isolated boundary points .

A point set without convergent points is called an isolated point set (scattered point set) because its induced topology must be a scattered topology .

A point set *S* (that is, *S **S'* ) without isolated points is called self-dense . Especially if *S* is self-dense and closed, then *S* is called a complete set . Because the necessary and sufficient conditions for *S* to be closed are: *S' **S* , so *S* The necessary and sufficient conditions for being a complete set are: *S* = *S* '._{}_{}

3.
Degree of Separation of Topological Spaces Countable Axioms

1. Topological spaces with different degrees of separation

[ *T *_{0} space ] If at least one of any two different points in a topological space *X* has a neighborhood that does not contain the other point, then *X* is called a *T *_{0} space . _{}

[ *T *_{1} space ] If any two different points in a topological space *X* must each have a neighborhood that does not contain the other point, then *X* is called a *T *_{1} space . _{}

*
The necessary and sufficient conditions for X* to be *T1* space are: Any set { _{x} } in *X* that contains only one point *x **is* a closed set .

[ *T *_{2} space - Hausdorff space ] If any two different points in the topological space *X* must have neighbors and no common points with each other, then *X* is called *T *_{2} space, also called separation space . _{}

[ Regular space ] Assuming that for any closed set *S* and any point *x *Ï *S* in the topological space *X* , there must be two open sets *U* and *V* , such that *U *ê *S* , *V **x* and *U* ∩ *V *1 *φ* , then *X* is called as Regular space . _{}

[ *T3*_{} space ] _{A} regular *T1 *_{space} is called _{a }*T3* space . _{}_{}

[ Normal space ] Assuming that for any two closed sets *A* and *B* with no common point in topological space *X* , there must be two open sets *U* and *V* such that *U *ê *A* , *V *ê *B* and *U* ∩ *V* = *φ* , then *X* is called as normal space .

[ *T4*_{} space ] _{A} normal *T1* space is called _{a }*T4* space _{.} _{}_{}

For example, the *n *_{-} dimensional real number space is *T4* space ._{}

The order of strength and weakness of the degree of separation is defined as follows:

normal regularity

*
T *_{4 }*T *_{3 }*T *_{2 }*T *_{1 }*T *_{0}* *_{}* *_{}* *_{}* *_{}

_{The} arrow means "must be" . For example , the *T4* space must be the *T3* space, and it must be the normal space . As for the normal and regular, the degree of separation cannot be compared, and they _{are }_{also} incomparable _{with }*T2* , *T1* and _{T0 }*.* ._{}_{}_{}* *_{}_{}

2. Countability

[ Neighborhood basis ] Assuming that *s* is a neighborhood family of a point *x* in the topological space , for any neighborhood *U of **x* , there must be *V **s* to make *V **U* hold, then *s* is called a neighborhood basis of *x* . * *_{}* *_{}* *

[ Covered family ] Assuming that the sum set of a family of point sets covers a set *S* , then the whole of this family of point sets is said to be a closed family of *S.*

[ The first countable space ] Assuming that any point in the topological space *X* has a countable neighborhood basis, then this space is called the first countable space ("the space that satisfies the first countable axiom") .

[ Lindeloff space ] Assuming that any closed-open set of any point set in a topological space has a countable sub-closed-cap family, then this space is called a Lindelof space .

[ Separable space ] Assuming that there is a set of countable points dense everywhere in a topological space *X* , then this space is called a separable space .

[ Second-countable space ] A space with a countable topological basis is called a second-countable space (a space that "satisfies the second-countable axiom") .

They have the following strong and weak relationships:

first countable

second countable separable

Lindelof

For example, the *n* -dimensional real number space *R ^{n}* is a second countable space, because {

4.
Limit and Continuity

[ Limit of transformation ] Suppose *f* is a transformation that transforms a set of points *A* in a topological space *X* into another topological space *Y.* Suppose also that *x *_{0} is a gathering point of *A.* If there is a point *y *_{0 in }*Y* , for *y* For any neighborhood *V of *_{0} , *x *_{0} has a neighborhood *G* such that _{}_{}_{}_{}

*f* ( ( *G* \ { *x *_{0} } ) ∩ *A* ) *V*_{}

Then call *y *_{0} the limit of *f* at *x *_{0 , denoted by}

_{}

Note that 1 ^{o} assumes that *y *_{0} is the limit of *f* at point *x *_{0 }*A'* , then there are only two cases, one is that *x *_{0} has a neighborhood *G* such that *f* ( ( *G* \ { *x *_{0} } )∩ *A* ) = { *y *_{0} } holds, otherwise, for any neighborhood *G of **x *_{0} , *y *_{0} is a cluster of *f* ( ( *G* \ { *x *_{0} } )∩ *A* ) ._{}_{}_{}_{}_{}_{}_{}

2 ^{o} in general, does not necessarily exist, and if it exists, it is not necessarily unique . But especially when *f* is a transformation that transforms *A* into a *T *_{2} space, either it does not exist, or it exists and is unique ._{}_{}_{}

[ Continuous transformation ] Suppose *f* is a transformation that transforms a point set *A* in a topological space into a topological space . Suppose *x *_{0} is an isolated point of *A or **x *_{0} is a gathering point of *A* and = *f* ( *x *_{0} ), then it is called *f* is continuous at *x *_{0} . _{}_{}_{}_{}_{}

If *f* ( *x* ) is continuous at every point *x **A* , then *f* is said to be continuous in *A* , or *f* is said to be a continuous transformation of *A.*_{}

Theorem The sufficient and necessary condition that the transformation *f* that transforms a point set *A* in a topological space into a topological space is a continuous transformation of *A is: Any relatively open image source of **f* ( *A* ) (that is, every relative open set in this relatively open set) The whole of the image source of a point) is a relatively open set of *A* (the "open" in the condition can be changed to "closed") .

[ The coarsest topology that makes a transformation continuous ] Assuming that a transformation *f* transforms a set *A* into a bearing point set *B* of a topological space , then take all relatively open image sources of *f* ( *A* ) as a set of *A* Topological subunit, we get a topology *τ* of *A.* This topology *τ* is the thickest topology that makes *f* continuous in *A.*

In particular, when *f* is a real function (or real functional) * defined in the set *A* , *f* can be regarded as a transformation that transforms *A* into *R *^{1} , so all forms are { *x* | *x **A* and *a* < *f* ( *x* ) A subset of < *b* } (where *a* and *b* are arbitrary real numbers) collectively breed the coarsest topology that makes *f* continuous .^{}_{}

[ Exploitation Theorem - Body Policy Theorem ] Assuming that *f* is a continuous bounded real function in a closed set *B* of normal space *X , for any **x **B* , holds, then there is a function *g* in *X* ( the set of bearing points of *X* ) is continuous, and holds for all *x **B* , , and for all points *x* in *X* . _{}_{}_{}_{}_{}

It is a generalization of the continuous function property of the following real variable function:

Assuming that *A* is a set of points in a topological space, *f *_{1} , *f *_{2} , ¼ are a list of continuous functions in *A* that converge uniformly to the function *f* (that is, for any positive number *e* , there exists a positive integer *N* such that | *f ** _{n}* (

[ Topological transformation and homeomorphism ] Assuming that *X* and *Y* are both topological spaces, *f* is a transformation that transforms the bearing point set of *X into the bearing point set of **Y* one-to-one . Under the transformation *f* , each opening in *X* The image of a set is an open set in *Y* , and the image source of each open set in Y is also *an open set in **X* , then *f* is called a topological transformation (homeomorphic transformation) that changes *X* to *Y , and **X* and *Y* are called Homeomorphic or topologically equivalent under *f* .

Theorem The sufficient and necessary condition for transforming the bearing point set of topological space *X* into the bearing point set of topological space *Y one-to-one into topological transformation is: **f* is reversible and continuous (that is, both *f* and *f *^{-1} are continuous transformations) .^{}

Five,
dot net

In the analysis of real variables, series, function series, function value series, etc. are common basic tools . These concepts can be summarized by a unified concept such as point series in topological space . However, for the limit theory of general topological spaces, point series The concept of column is too narrow and should be extended to the concept of dot net, the role of dot net is equivalent to that of dot column in real variable analysis .

[ Summary set ] Assume that *Q* is a set with a magnitude relation < in *Q* , and satisfy the condition: ( i ) For any *p **Q* and *q **Q* , the formula *p* < *q* ,
*p* = *q* ,
*q* < *p* has one and only one holds; ( ii ) if *p* , *q* , *r* all belong to *Q* , and *p* < *q* and *q* < *r* hold, then *p* < *r* ; ( iii ) for any *p **Q* and _{}_{}_{}*q **Q* , there is *r **Q* such that *p* < *r* and
*q* < *r* are all established . Then *Q* is called a summary set ._{}_{}

As you can see from the definition, the aggregate set is the branch set that satisfies the condition ( iii ) .

[ dot net ] The transformation *f* of a summary set *Q* into a topological space *X* is called a dot net in *X* , and the image *f* ( *q ) of a point **q **Q* is usually recorded as *x ** _{q}* , so <

Especially when *Q* is the set of finite ordinal numbers *ω* or the set of positive integers, the dot net < *x _{q}* |

[ Two definitions of dot net limit[5191] ]

1° Assume *Q* is a collection set, regard *Q* as a scattered space . Take any thing that does not belong to *Q* (such as *Q* itself) and record it as ∞, which is called the infinity of *Q* (the ultimate) . Take all the things in ∞ and *Q* The totality of elements *q* larger than some element *p* (that is, { ∞ } ∪ { *q* | *q **Q* and *q* > *p* } ) is defined as an open set in *Q* ∪ { ∞ } . Under the above specification, in *Q* ∪ { ∞ } propagate a topology . Under this topology, *Q*_{}*becomes a set of points in a topological space, and ∞ is the only foci of **Q.*

A point net < *x ** _{q}* |

If a limit of a point net exists, the point net is said to converge to this limit .

2° Assuming that < *x _{q}* |

_{}

means that for any neighborhood *V of **a* , there is always a *p **Q* such that for all *q* > *p* , *x *_{q}* V* holds ._{}_{}_{}

This is more formally consistent with the definition of the usual point sequence limit .

[ Subnets and Aggregation Limits ] Assuming that *Q *_{1} is an unbounded subset of the aggregated set *Q* (that is , no element in Q can be greater than *all* elements of *Q *_{1 ), then }*Q *_{1} is called the co-terminal of *Q* Subsummary set . _{}_{}_{}

The reason for this name is that *Q *_{1} must also be an aggregate set, and in *Q* ∪ { ∞ } , the ultimate ∞ is also the only aggregation point of *Q *_{1} .

Assuming that < *x _{q}* |

The limit of the subnets of a point net is called an aggregate limit of the point net .

Theorem 1 In a topological space, a necessary and sufficient condition for a point *x *_{0} to be a convergent point of a point set *A is: There is a point network in **A* \ { *x *_{0} } that converges to *x *_{0} ._{}_{}

Infer that in a first countable space, the necessary and sufficient condition for a point *x *_{0} to be the aggregation point of point set *A* is: *A* \ { *x *_{0} } has a point sequence that converges to *x *_{0} . _{}_{}_{}

Theorem 2 assumes that *A* is a subset in a topological space, *x *_{0} is the aggregation point of *A , and **f* is the transformation that transforms *A* into a topological space, then the necessary and sufficient conditions are: for all in *A* \ { *x *_{0} } Dot nets that converge to *x *_{0}_{} _{}_{}

< *x _{p}* |

[ The point-to-point convergence topology of the transformation family ] The totality of transformations that transform a set *A* into a topological space *Y* is the overlapping set ^{A}* Y* . ^{A}* Y* can actually be regarded as a direct product *Y ** _{x}* , where each

Since *Y* is a topological space, * ^{A} Y* or

The theorem assumes that *A* is a set, *Y* is a topological space, then compared with other topologies of ^{A}* Y* , the point-to-point convergence topology is characterized by: * ^{A}* sufficient and necessary condition for the convergence of any point network <

* When* A* is a family of functions, it is customary to refer to real functions defined in* A* as real functionals.

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