**Chapter 21 Set Theory and General Topology**

Set is the most basic concept of modern mathematics . This chapter first introduces the axiom system of set theory . However, this system has recently been proved to be incomplete , so the necessary explanation is given for the starting point taken here ( see § 1 , the last of I ) .Secondly, it introduces the main content of set theory itself-the theory of ordinal and cardinality .

Another main content of this chapter is general topology . Here we focus on several special topological spaces and point sets that are particularly important to mathematical analysis - scale space ( with a consistent structure ) , compact set, connective set, and a combination of the former two The point-to-point convergent topology, uniformly convergent topology, and compact - open topology of the transformation family are discussed . Finally, the concepts of manifolds and differential manifolds and several basic existence theorems are introduced . The knowledge of algebraic topology is not covered in this chapter (the only The exception is the notion of "simple" in § 6 ), and furthermore, the introduction of differential manifolds does not involve differential geometry (such as tangent spaces) .

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The definition of a set

1. Classical Definition of Sets

[ Set and element ] The whole of some things is called a set , and each of these things is called an element of this set ( or in this set ) .

If there is only one thing , and this thing is supposed to be denoted *a* , then the whole of this thing is called the set { *a* }, and *a* is the only element of { *a* } .

If a certain thing does not exist , the whole of such things is said to be an empty set . It is stipulated that any empty set is just the same set , denoted by *φ* . Nothing is an element of *φ* .

Each set is a thing .

[ belongs to and contains ] Suppose *a* is an element of set *A* , denoted as

*a **A* or *A **a*_{}_{}

" " is read as "belongs to" and " " is read as "contains" . Assuming that *a* is not an element of *A , write*_{}_{}

_{}or_{}

" " is read as "does not belong to" and " " is read as "does not contain" ._{}_{}

[ defined note ]

1° { *a* } and *a* are generally different concepts , such as { *φ* } has a unique element *φ* , but *φ* has no elements .

2°
and are logically negated ( not ) of each other , in other words , suppose *a* is a thing and *A* is a set , then_{}_{}

*a **A* and *a **A*_{}_{}

Neither can be established , nor both can be established .

3° Assuming that *A* and *B* are both sets , if any thing belongs to *A* and must belong to *B* , and belongs to *B* must also belong to *A* , then *A* and *B* are the same set , or two sets *A* and *B* are equal , denoted as *A* = *B.* _

[ Example of set ] Suppose there are some things , all written out as *a* , *b* , *c* , ... , then by definition , their whole is a set , and this set can be written as { *a* , *b* , *c* , ... }. Element notation The order and repetition are irrelevant , such as { *a* , *b* }={ *b* , *a* }={ *a* , *b* , *a* }.

By definition , *φ* is a set , and a set is a thing , so the following things are sets :* *

{ *φ* },{{ *φ* }, *φ* },{{{{ *φ* }},{ *φ* }},{ *φ* }}

For another example , zero and positive integers can be defined as follows :

0 = *φ*

1 = { 0 } = { *φ* }

2 = { 0 , 1 } = { *φ* , { *φ* }}

3 = { 0 , 1 , 2 } = { *φ* ,{ *φ* },{ *φ* ,{ *φ* }}}

4 = { 0 , 1 , 2 , 3 }

…………

[ Family ] Family is a synonym for set . In some cases , such as when the elements of a set *A* are all sets , in order to avoid confusion , *A* is also called a family or a set family .

Although in modern set-theoretic models , the elements of any set are sets ( because "things" that are not sets are not considered ) , sometimes the term "family" is used to express this more clearly .

Family is also sometimes used as a quantifier . For example, all sets belonging to a set family are said to be "a family set" .

2. Russell Weird

The example above has been used to illustrate how to represent a set by enumerating elements . But when all the elements of a set cannot be enumerated , how should the set be represented ? is "the whole of all things that satisfy a certain condition" . If the sentence "a certain thing *x* satisfies a certain condition" is expressed as a logical formula *p* ( *x* ), then according to this customary notation , a set It can be written as { *x* | *p* ( *x* )} or { *x* : *p* ( *x* )} ( the totality of all *x* that makes *p* ( *x* ) true ). *x* , *p**( **x* ) and ( not *p* ( *x* ), that is , the negation of *p* ( *x* ) ) have one and only one true , then this notation is not a problem . But it is not the case in practice . Take the famous Russell's anomaly when Example :_{}

Suppose . If *z* is a set , then *z* is also a thing , so *z **z* and *z **z* cannot both hold . Suppose *z **z* , then *z* should satisfy the stated condition *x **x* , so *z **z* , contradicting itself . Assuming *z **z* , then *z* already satisfies the stated condition *x **x* , so *z **z* , which contradicts itself . This is called Russell's weirdness ._{}_{}_{}_{}_{}_{}_{}_{}_{}

According to the definition's comment , *z* is not a set . So the Russell weirdness is actually caused by the false assumption that " *z* is a set" . Apart from this formal logical reason, the Russell weirdness can be explained in more depth, but there is a fundamental The problem of is not easy to solve. Since { *x|x **x* } is not a set, can the other { *x|p* ( *x* )} be counted as a set ?_{}_{}

To answer this question , the concept of sets must be refined further , so the axiom system is introduced below .

3. ZFC axiom system and BNG axiom system

At present, there are two forms of the axiomatic system of set theory , one is the Zemolo - Frankel - Korean form , referred to as ZFC ; the other is the Bernes - Neumann - Gedel form , referred to as BNG . ZFC axioms are used here . system .

ZFC includes nine axioms ( three are clearly included in the definitions of the preceding sets and in the annotations of the definitions ), which are

[ Axiom of extension ] is the annotation of the definition . _{}

[ Empty Set Axiom ] There exists a set that contains no elements .

[ Axiom of Unordered Pairs ] For anything *x* and *y* , there exists a set { *x* , *y* }, where the only elements of { *x* , *y* } are *x* and *y* .

[ Regular Axiom ] Any non-empty set *A* must contain an element *a* , and any element of *A* is not an element of *a* .

It can be known from the axiom of regularity that for any set *a* , *a* and { *a* } are different . This is because if *a* = { *a* }, then { *a* } does not conform to the axiom of regularity .

The remaining five axioms of ZFC are the axiom of substitution ( section , 2 ), the axiom of power sets of squares, the axiom of sum sets ( section , 3 ), the axiom of infinity ( § 2 , 3 ), the axiom of choice ( § 2 , 4 ). They are They are explained in detail in their respective sections . In general , these axioms specify what sets are in a more precise form . But this system of axioms cannot prove that it does not contradict itself , and it does not include all the sets necessary for set theory. are stipulated in ( § 2 , 6 ). This system therefore fails to replace the classical definition of set . The following starting points will be used later : ( i )Assume that the sets specified by these five axioms conform to the classical definition of sets and the annotations of the definitions . ( ii ) Except for sets whose elements can be enumerated in their entirety , only the sets specified by the above axioms are considered .

2.
Transformation · General notation for sets · Sets of labels

[ Ordered pair ] Assuming that *x* and *y* are things , then

< *x* , *y>* = {{ *x* , 1 },{ *y* , 2 }}

It is called an ordered pair formed by *x* and *y* , and *x* and *y* are called the first and second coordinates of < *x* , *y* > , respectively .

Ordered pairs are said for unordered pairs . It can be seen that the necessary and sufficient conditions for < *x', y' >* = < *x, y>* are : *x'* = *x* and *y'* = *y* , and the unordered pairs follow the elements one after the other The order doesn't matter .

[ Replacement axiom ] Assuming that *X* is a set , if for each *x **X* as the first coordinate , there is one and only one *y* and *x* forming an ordered pair < *x* , *y >* , then all the ordered pairs of the first coordinate The whole of the two-coordinate *y* is a set *Y* . _{}

Considering each < *x* , *y>* as the second coordinate of an ordered pair < *x* , < *x* , *y>>* , and applying the axiom of substitution again , we can see the totality of all such ordered pairs < *x* , *y>* Also a set .

[ transformation ( mapping ) image source ( original image ) image ] Assuming that X *is* a set , if for each *x **X* as the first coordinate , there is one and only one *y* and *x* form an ordered pair < *x* , *y >* , the whole of its second coordinate *y* is denoted as *Y* ( is a set ), then the whole of all such ordered pairs *< **x, y>* is a set *f* , then *f* is called the transformation from *X* to *Y* ( map ), abbreviated _{}*f* is transformed ( mapped ), *X* is called the image source ( original ) of *Y* under transformation *f , and **Y* is called the image of *X under transformation **f* , denoted as *Y* = *f* ( *X* ).

In general , assuming < *x* , *y > **f* , then denoted as_{}

*y* = *f* ( *x* )

*x is* called the image source of *y* under transformation *f , and **y* is called the image of *x under transformation **f* .

[ One-to-one transformation and inverse transformation ] By definition , each image source of a transformation has only one image ( uniqueness ), but an image does not necessarily have only one image source . If special, each image also has only one image source , then *f* is said to be a one-to-one transformation . Under a one-to-one transformation *f* , a transformation that changes *Y* to *X* can be obtained , which is called the inverse transformation of *f* . If *f* ( *x* ) = *y* , then ( *y* ) = *x* . _{}_{}_{}

[ General representation of sets and set of labels ] Suppose there is a one-to-one transformation that converts a set *H* into *X* , then *X* is a set , if the image of each image source *h* ( *H* ) is denoted by *x ** _{h}* (

*X*
= { *x _{h} * |

Then *H* is called the label set of *X* , and each *h* is called the label of *x ** _{h}* .

Conversely , a set has a label set . Because at least it can be regarded as its own label set . Therefore , the expression ( 1 ) is generally applicable . When applying this notation in the future, it is not necessary to specify *H* is a label set , as long as it is stipulated that the symbol written in the *H* position must be a label set .

3.
The set specified by the axiomatic system

[ Subset ] Assuming that *A* and *B* are both sets , and each element of *B* is an element of *A* , then *B* is called a subset of *A* , and denoted as *B **A* or *A **B.* " " is read as "contained in" or "Covered in", " " is read as "includes" or "covers up" . _{}_{}_{}_{}

For any set *A* , *B* and *C* have

1°
*A **A* ( Reflexive Law )_{}

2°
From *A **B* , *B **A* , it can be deduced that *A=B* ( antisymmetric law )_{}_{}* *

3°
If *A **B* , *B **C* , then *A **C* ( transitive law )_{}_{}_{}

Suppose *B **A* but *B **A* ( *B* = *A* does not hold ), then call *B* a proper subset of A *,* denoted by ( *B **A* )._{}_{}_{}

Specifies that the empty set is a subset of any set ._{}

[ Transformation transformation ] Suppose a transformation *f* transforms a set *X* into a subset of the upper set *Y* , then *f* is called the transformation that changes *X* into *Y , and **f* for short is transformed ( mapped ). The transformation is A special case of change .

[ Division axiom and characteristic function ] Assuming that there is a transformation *f* that changes a set *X* into { 0 , 1 }, then the totality of all image sources of 1 is a subset *X **'* of X , and *f is* called the characteristic function of *X'* .

The division axiom is the conclusion of the substitution axiom , because if the totality of the image sources of 1 is *φ* , then *φ* is of course a subset of *X* , otherwise 1 has at least one image source *x *_{0}* X* , making a transformation_{}_{}

_{}

Then *g* ( *X* ) = *X'* , so *X* is the set .

The corollary assumes that *X* is the set , and for each *x **X* , the arguments *p* ( *x* ) and ( the negation of *p* ( *x* ) ) must hold one and only one , then { *x* | *x **X* and *p* ( *x* )} is a set . _{}_{}_{}

[ Difference set and complement set ] Assuming that *A* and *B* are both sets , then the whole of all elements belonging to *A* but not *B* is a set ( inference by the axiom of division ), and the difference set called *A* and *B* is recorded as *A* \ *B.* _

In particular , when *B is **A* , *A* \ *B* is called the complement of *B* in *A.*_{}

[ Axiom of Square Power Sets ] The totality of all subsets of a set *A* is a set , denoted as *A* -square power set . _{}

One-to-one can be changed to "the set of all transformations that transform *A* into 2 = { 0 , 1 } ", so the latter is also a set, and this set and the *A* -square power set can be used as each other's label sets . In the future , they are often regarded as the same set , that is , a subset of A is confused with one of its characteristic *functions* ._{}

[ Axiom of Sum Set ( Union ) and Sum Set ] Assuming that { *A ** _{h}* |

When all the sets of a family are *A* , *B* , *C* , ... , the sum set of this family can be written as

*
A* ∪ *B* ∪ *C* ∪ …

Example { 1 , 2 , 3 } ∪ { 0 , 2 , 4 } ∪ { 2 , 1 }={ 0 , 1 , 2 , 3 , 4 }

[ Common set ( intersection ) ] Assuming that { *A ** _{h}* |

When all sets of a family are *A* , *B* , *C* , ... , the general set of this family can be written as

*A* ∩ *B* ∩ *C* ∩ …

Example ** ** { 1 , 2 , 3 } ∩ { 0 , 2 , 4 } ∩ { 2 , 1 }={ 2 }

[ Direct product ( Cartesian product ) ] Assuming that *A* = { *x ** _{h}* |

{ < *x _{h}* ,

is a set called the direct product of *A* and *B* , denoted *A **B* ._{}

Direct product existence is the conclusion of the substitution axiom and the sum set axiom . Since for any *h **H* and *k **K* , { < *x ** _{h}* ,

Assuming that { *A _{h}* |

< *x _{h}* |

is called an ordered group obtained by a selection transformation ( §2,4 ) .

Replace each *x _{h}*

< *x ^{' }_{h}* |

This can also be seen as an ordered group obtained by a selection transformation .

The totality of all such ordered groups is a set, which is called the direct product of a family of sets *A _{h}* (

* When H* = 2 , it _{}is *A B. _{}*

[ Overlapping Sets ] Assuming that *A* and *B* are both sets , then by the definition of transformation , each transformation *f* that changes *A* into *B* is a subset of *A **B* , so *f* . By the axiom of division , all transformations that change *A* into *B* are The whole of the transformation *f* { *f* | *f* and *f* transforms *A* into *B* } is a set , called the overlapping set that superimposes *A* on *B* , denoted ^{A}* B* . _{}_{}_{}_{}_{}^{}

Obviously , * ^{
A} B* . On the other hand , especially when

[ Operation Law of Sets ] Assuming that *A* , *B* , and *C* are all sets , then

Commutative law *A* ∪ *B* = *B* ∪ *A* , *A* ∩ *B* = *B* ∩ *A*

Associativity *A* ∪ ( *B* ∪ *C* ) = ( *Α* ∪ *B* ) ∪ *C*

*A* ∩ ( *Β* ∩ *C* ) =( *Α* ∩ *B* ) ∩ *C*

The distributive law *A* ∩ ( *B* ∪ *C* ) = ( *A* ∩ *B* ) ∪ ( *A* ∩ *C* )

*A* ∪ ( *B* ∩ *C* ) =( *A* ∪ *B* ) ∩ ( *A* ∪ *C* )

De Morgan 's Law _ _ _ _

*
C* \( *A* ∪ *B* ) = ( *C* \ *A* ) ∩ ( *C* \ *B* )

*
C* \( *A* ∩ *B* ) = ( *C* \ *A* ) ∪ ( *C* \ *B* )

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