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 made on the starting point adopted here (see § 1 , the last ). Next, introduce the main content of the set theory itself-ordinal number and cardinal number theory.

Another main content of this chapter is general topology. Here we focus on several special topological spaces and point sets that are particularly important for mathematical analysis - scale space (with consistent structure), compact set, connection set, and a combination of the former two The point-to-point convergence topology, uniform convergence topology, 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 included in this chapter (the only The exception is the concept of "single connection" mentioned in §6 ). In addition, the introduction of differential manifolds does not involve differential geometric structures (such as tangent spaces).


§1 Set (collection)


one,          Set definition


    1. Classical definition of set

[ Set and Elements ] The totality 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 certain thing, and this thing is assumed to be recorded as a , then the whole of such things is called the set ( a ), and a is the only element of ( a ).

If a certain thing does not exist, it is said that the whole of such things is 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 episode is a thing.

[ Belonging to and Containing ] Assuming that a is an element of set A , denote it as  

a A or A a

" " Is pronounced as "belongs to" and " " is pronounced as "contains". Assuming that a is not an element of A , write it as


" " Is pronounced as "does not belong to" and " " is pronounced as "does not contain ".

    [ Defined comment ]

1°   { a } and a are generally different concepts. For example, { φ } has a unique element φ , but φ has no elements.

2°   And logically are negative (not) to each other. In other words, assuming that a is a thing and A is a set, then

a A and a A

It can't all be true, and it can't all be true.

3°   It assumed that A and B are set, if any one thing belonging to A must also belong to B , belonging to B must also belong to A , then A and B are the same set, or said two sets A and B are equal, denoted by A = B .

    [ Set example ] Suppose there are some things, all written out as a , b , c , … , then by definition, all of them are a set, this set can be written as { a , b , c ,...}. Element symbol The order and repetition of 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 all 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 ] A family is a synonym for a 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 the modern set theory model, the elements of any set are sets (because non-set "things" are not considered), sometimes the term "family" can be used to express more clearly.

    Clan is sometimes used as a quantifier. For example, all the ji belonging to a ji family are called "a family ji".

2. Russell weird

    The above has used examples to illustrate how to use the method of enumerating elements to represent a set. But when all elements of a set cannot be enumerated, how should the set be represented? In the early days of the development of set theory, the popular habit was to say a set Cheng is “the totality of all things that satisfy a certain condition”. If the sentence “something x satisfies a certain condition” is expressed as a logical formula p ( x ), then according to the said customary notation, a set can be written as { the X- | the p- ( the X- )} or { the X- : the p- ( the X- )} (so that all the p- ( the X- ) established the X- . plenary) generally tend to believe that as long as the said condition is clear, that is, for any x , p ( x ) and (not p ( x ), which is the negation of p ( x )) have one and only one valid, then this notation is okay. But in fact it is not the case. The following famous Russell weird as an example:

    Suppose . If z is a set, then z is also a thing, so z z and z z cannot both be true. Suppose z z , then z should satisfy the condition x x , so z z , contradicts itself. Suppose z z , then z already satisfies the said condition x x , so z z is contradictory. This is called Russell's weirdness.

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

    In order to answer this question, the concept of sets must be refined, so the axiom system is introduced below.

    3. ZFC axiom system and BNG axiom system 

    At present, the set theory axiom system has two forms, one is the Zermelo-Frankel-Ke very form, referred to as ZFC ; the other is the Bernese-Neumann-Gödel form, referred to as BNG . Here, ZFC axioms are used. system.

    ZFC includes nine axioms (there are three obviously included in the definition of the previous set and the comments on the definition), they are

    [ Extension Axiom ] is the comment of the definition .

    [ Empty Set Axiom ] There is a set that does not contain any elements.

    [ Disordered Pair Axiom ] For any thing x and y , there exists a set { x , y }, 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 canonical axiom that for any set a , a and { a } are different. This is because if a = { a }, then { a } does not conform to the canonical axiom.

    The remaining five axioms of ZFC are substitution axioms (this section, two), power set axioms, sum set axioms (this section, three), infinite axioms ( §2 , three), and choice axioms ( §2 , four). They are explained in detail in the respective joints. In general, these axioms specify which sets are in a more precise form. But this axiom system cannot prove that it is not contradictory, and it does not integrate all the sets necessary for set theory. All are stipulated ( §2 , 6). Therefore, this system failed to successfully replace the classical definition of set. The following starting points will be adopted later: ( i ) Assume that the set stipulated by these five axioms conforms to the classical definition of the previous set ( Ii ) Except for the set in which all elements can be enumerated, only the set specified by the above axiom is considered.


two,          Transform ·General notation of set·label set


    [ Ordered Pair ] Assuming that x and y are both 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 for disordered pairs. It can be seen that the necessary and sufficient conditions for < x',y'> = < x,y> are: x' = x and y' = y , and disordered pairs follow the elements The order is irrelevant.

    [ 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 form an ordered pair < x , y> , then the first of all such ordered pairs second coordinate y is a whole set of the y .

    The each < X , Y> viewed as an ordered pair < X , < X , Y >> second coordinate, again applied axiom Alternatively, you can see all such ordered pairs < X , Y> entire It is also a set.

    [ Transformation (mapping) • Image source (original image) • Image ] Assuming that X is a set, if each x X is used as the first coordinate, there is one and only one y and x form an ordered pair < x , y > , a second coordinate y all referred to as a Y (a set), then all such ordered pairs < X, y> of a whole set of f , said time f of the X variable to Y on conversion (mapping), referred to as f , the (reflected on) a variable X referred transform f the Y of the source image (primary image), Y referred transform f the X image, denoted by Y = f ( X ).

    Generally, suppose < x , y> f , then write it as

y = f ( x )

x referred transform f the y image source, y referred transform f at x elephant.

    [ One-to-one transformation and inverse transformation ] By definition, each image source of a transformation has only one image (singularity), but an image does not necessarily have only one image source. If in particular, each image source also has only one image source. , then said f is a one to one transformation in a one to one transformation f under the get a Y on the variable X conversion , called f inverse transform. If f ( X ) = Y , then ( Y ) = x .

    [ General Representation of Sets and Label Sets ] Assuming that there is a one-to-one transformation to change a set H to X , then X is a set. If the image of each image source h ( H ) is written as x h ( X ), and write X as

X ={ x h | h H } ( 1 )                                

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 notation of formula ( 1 ) is universally applicable. It is not necessary to explain H when applying this kind of notation in the future. It is a label set, as long as it is stipulated that the H position in this kind of mark must be a label set.


three,          Set of axiom systems


    [ Subset ] Assuming that A and B are both sets, and each element of B is an element of A , then call B a subset of A , denoted as B A or A B. " " is pronounced "contained in" or "Be covered by", " " is pronounced "contained" or "covered 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 (anti-symmetric law)  

3°   If A B , B C , then A C (transmission law)       

    Suppose B A but B A ( B = A does not hold), then call B a proper subset of A , denoted as ( B A ).

    Specifies that the empty set is a subset of any set.

    [ Change-in transformation ] Suppose that a transformation f changes a set X into a subset of the upper set Y , then f is the transformation that changes X into Y , and f is the change-in (reflected) for short . The change-up is Special circumstances of change.

    [ Division axiom eigenfunction ] Consider a transformation f to a set of X becomes into { 0 , 1 }, then 1 all of the entire image source is X a subset of X ' , f called X' characteristic function.

    The division axiom is the conclusion of the replacement axiom, because if the image sources of 1 are all φ , then φ is of course a subset of X , otherwise 1 has at least one image source x 0 X , make a transformation


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

    The corollary assumes that X is a set, and for each x X , the argument p ( x ) and ( The negation of p ( x )) must have one and only one true, then ( x | x X and p ( x )) is a set.  

    [ Difference Set and Complementary Set ] Assuming that A and B are both sets, then the totality of all the elements belonging to A but not belonging to B is a set (inference from the division axiom), which is called the difference set of A and B and denoted as A \ B .

    In particular, when B A time, A \ B referred to as B in A over the set.

    [ Axiom of Power Set ] The totality of all subsets of a set A is a set, denoted as , called A power set.

    Can be one becomes the "All the A variant into 2 = { 0 , 1 } is transformed whole" go, the latter is a set, and sets the A side of the power set of the label sets as each other mutually. in the future, tend to see them as the same set, that is, the a a sub-set of a feature function is saying confuse.

    [ Sum Set (Union) and Sum Set Axiom ] Assuming that { A h | h H } is a set family, then { x | exists a A h x } is a set, which is called the sum set (union) of this family set , Recorded as .

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

                             A ∪ B ∪ C ∪…

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

    [ General set (intersection) ] Assuming that { A h | h H } is a set family, then { x | all A h x } is a set, which is called the general set (intersection) of this family set, denoted by . The existence of a general set is the conclusion of the division axiom.

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

                         A ∩ B ∩ C ∩...

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

    [ Direct product (Cartesian product) ] Suppose A = { x h | h H }, B = { y k | k K }, then

                     { < x h , y k > | x h A and y k B }

Is a collection, which is called A and B direct product, denoted by A B .

    Direct product existence is the conclusion of replacement axioms and sum set axioms. Because for any h H and k K , { < x h , y k > } is a set, by the replacement axiom, {{ < x h , y k > }| H H } is a set of family, there is set, and {. C K | K K } is a set of a group, it is also present and set , which is a B .

    Assuming that { A h | h H } is a set family, where each A h  φ, then by the choice axiom ( §2 , 1), for each h H, one x h A h can be obtained , and a set is obtained by the substitution axiom 

< x h | h H>= {{ x h ,h }| h H }

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

    Replace each x h A h with an x ' h A h , then another set is obtained by the substitution axiom 

< x ' h | h H>= {{ x ' h ,h }| h 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 set A h ( h H ), denoted by .

    When H = 2 , it is A B.

    [ Bundle set ] assumed that A and B are set, then the transformation defined by each of the A variant into B transformation f is A B subset, so f . Divided by the axiom, all the A variant into B of transformation f entire { f | f and f the a variant into B } is a set, called the a laminated B on stack set, denoted as a B .

    Obviously, A B. On the other hand, especially when B = 2 = ( 0 , 1 ), A 2 is both a power set and a stack set.

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

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

    Associative law A ∪ ( B ∪ C ) = ( Α ∪ B ) ∪ C       

A ∩ ( Β ∩ C ) = ( Α ∩ B ) ∩ C

    Distributive law A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C )       

A ∪ ( B ∩ C ) = ( A ∪ B ) ∩ ( A ∪ C )

    Germany  Morgan ( De Morgan ) law

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

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