A.1.4. Finite Set, Countable Set and Uncountable Set
Definition 1.4.1
A and B are two sets. If there exists a 1-1 surjective mapping from A to B, A and B are called equinumerous.
Any set that is not equinumerous to its proper subsets is a finite set.
A set that is equinumerous to the set N of all natural numbers is a countable set.
An infinite set that is not equinumerous to the set N of all natural numbers is an uncountable set.
Theorem 1.4.1 (Bernstein)
If A and the subset of B are equinumerous, and B and the subset of A are also equinumerous, A and B are equinumerous.
A.7.2. Closure Operation and Closure Space
The concept of closure operation that we previously introduced is under the order theory sense. The terms of closure operation, closure space and related properties that we will introduce below have the topologic sense, especially under E. Cech sense, i.e., based on set theory and always assuming that there does not appear paradox (
Cech, 1966).
Definition 7.2.1
is a domain. If mapping
satisfies the following three axioms, where
is a power set of
,
(cl1)
(cl2) ,
then,
is called a closure operation on
. Correspondingly, two-tuples
is a closure space, and
is a
closure of subset
. If not causing confusion, the closure
of subset
is denoted by
.
Proposition 7.2.1
If
is a closure space, then
(1)
(3) For any family
of subsets of
, have
Definition 7.2.2
is a set composed by all closure operations on
, i.e.,
=
is a closure operation on
}. Define a binary relation
on
as
If
holds, then closure operation
is said to be coarser than
. Equivalently,
is said to be finer than
.
Theorem 7.2.1
Binary relation
is a semi-order relation on
.
has a greatest element
and a least element
. That is, for
, if
, then
, otherwise
; and
. Furthermore, for any subset
of
and
, we have
, i.e.,
is order complete with respect to
.
Definition 7.2.2
is a closure space. Mapping
, induced by closure operation
, is called an interior operation, denoted by
. Its definition is as follows
Correspondingly,
is called
interior of
, or simply interior.
Proposition 7.2.3
is a closure space. If
is defined by Definition 7.2.2, then
(int1)
(int2)
,
Assume that
satisfies axioms
int1∼
int3. Define an operation
as follows
It can be proved that
is a closure operation on
and
. If
is a set of mappings
on
that satisfy axioms
int1∼
int3, then there exists one-one correspondence between
and
. Or a closure operation and an interior operation are dual.
Definition 7.2.3
is a closure space.
is a dual interior operation of
. For
, if
, then
is called a close set. If
, or equivalently,
, then
is called an open set.
Proposition 7.2.4
is a closure space.
is a dual interior operation of
. We have
(1)
(2) For
and
, if
, then
.
(3) For any family
of subsets of
, have
Definition 7.2.4
A topological closure operation on
is a closure operation
that satisfies the following condition
If
is a topological closure operation, then closure space
is a topological space.
Proposition 7.2.5
If
is a closure space, then each condition shown below is the necessary and sufficient condition that
is a topological space.
(1) The closure of each subset is a close set
(2) The interior of each subset is an open set
(3) The closure of each subset equals to the intersection of all close sets that include the subset
(4) The interior of each subset equals to the union of all open sets that include the subset.
Theorem 7.2.2
Assume that
is a family of subsets of set
that satisfies the following conditions
(o1) ,
(o2) ,
, i.e.,
is closed for any union operation
(o3) ,
, i.e.,
is closed for finite intersection operation.
Let
is a closure operation on
and the set composed by all open sets of
is just
.
Then, there just exists a topological closure operation
on
such that
is the roughest element on
.
Theorem 7.2.3
Assume that
is a family of subsets of set
that satisfies the following conditions
(c1) ,
(c2) ,
, i.e.,
is closed for any union operation
(c3) ,
, i.e.,
is closed for finite intersection operation.
Then, there just exists a topological closure operation
on
such that
is just a set that composed by all close sets on
.
Using open set as a language to describe topology, axioms (
o1) ∼ (
o3) are used. However, conditions (
cl1) ∼ (
cl4) are called axioms of Kuratowski closure operator. Kuratowski closure operator, interior operator that satisfies axioms (
int1) ∼ (
int3) and
(int4):
, open set and neighborhood system are equivalent tools for describing topology. For describing non-topologic closure spaces, only closure operations, interior operations and neighborhood systems can be used, but open set or close set cannot be used as a language directly. In some sense, closure spaces are more common than topologic spaces. We will discuss continuity, connectivity and how to construct a new closure space from a known one below.
A closure operation
on a domain set
is defined as a mapping from
to itself, where domain
and codomain
. Closure operation
is completely defined by binary relation
, i.e.,
and
,
. Obviously, we have
.
Compared to
, relation
more clearly embodies the intuitive meaning of closure operation, i.e., what points are proximal to what sets. Naturally, the intuitive meaning of continuous mappings is the mapping that remains the ‘
is proximal to subset
’ relation.
Definition 7.2.5
is a mapping from closure space
to closure space
. For
and
, if
, have
holds, then
is called continuous at
. If
is continuous at any
, then
is called continuous.
Theorem 7.2.4
is a mapping from closure space
to closure space
. The following statements are equivalent.
(1) f is a continuous mapping
(2) For
,
holds.
(3) For
,
holds.
Definition 7.2.6
is an 1-1 correspondence (bijective mapping) from closure space
to closure space
. Both
and
are continuous mappings. Then,
is called a homeomorphous mapping from
to
, or
is a homeomorph of
.
Definition 7.2.7
If there exists a homeomorphous mapping from closure space
to
, then
and
are called homeomorphous closure spaces.
Definition 7.2.8
If a closure space
has property
such that all spaces that homeomorphous to
have the property, then
is called the topological property.
Obviously, the homeomorphous relation is an equivalent relation on the set composed by all closure spaces.
Definition 7.2.9
is a closure space. For
, if there exist subsets
and
on
such that
, and if
, then
or
, then
is called a connected subset of
.
Definition 7.2.10
is a continuous mapping from closure space
to closure space
. If
is a connected subset, then
is a connected subset on
.
Below we will discuss how to generate a new closure operation from a known closure operation, or a set of closure operations. Two generated approaches are discussed, the generated projectively and generated inductively. The product topology and quotient topology discussed in point topology are special cases of the above two generated approaches in closure operation.
Definition 7.2.11
is a set of closure spaces. For any
, the closure operation on
generated inductively by mapping
is defined as follows
The above closure operation is the finest one among all closure operations that make
continuous.
The closure operation on
generated inductively by a set
of mappings is defined as follows
The above closure operation is the finest one among all closure operations that make each
,
continuous.
Proposition 7.2.6
is a closure space.
is an equivalence relation on
, and its corresponding quotient set is
, where
,
. The closure operation
generated
inductively by
is defined as a quotient closure operation on
. And for
,
Definition 7.2.12
is a set of closure spaces. For any
, the closure operation on
generated projectively by
is defined as follows
The above closure operation is the coarsest one among all closure operations that make
continuous. The closure operation on
generated projectively by a set
of mappings is defined by
. It is the coarsest one among all closure operations that make each
,
continuous.
Note that
is not necessarily the
. And the latter is not necessarily a closure operation, unless a set
of closure spaces satisfies a certain condition (
Cech, 1966).
A.7.3. Closure Operations Defined by Different Axioms
Two forms of closure that we mentioned previously are denotes by closure operator and closure operation, respectively. The former is under order theory sense and the latter is under topologic sense. In fact, the term of closure does not have a uniform definition. In different documents it might have different meanings. We introduce different definitions of closure, quasi-discrete closure space, Allexandroff topology, etc. below.
is a domain. Assume that
is a given mapping. For
,
is called the closure of subset
.
is called the most general closure space.
is a dual mapping of
, i.e.,
.
is called the interior of subset
. For convenience, for
, the following axioms are introduced (
Table 7.3.1).
(CL0)
(CL2)
(CL3)
(CL4)
(CL5) for any family
of subsets on
,
.
where, (
CL1)+(
CL3) are equivalent to axiom (
CL3)′:
.
Table 7.3.1
|
(CL0) |
(CL1) |
(CL2) |
(CL3) |
(CL4) |
(CL5) |
Neighborhood space |
♦ |
♦ |
♦ |
|
|
|
Closure space |
♦ |
♦ |
♦ |
|
♦ |
|
Smith space |
♦ |
♦ |
|
♦ |
|
|
Cech closure space |
♦ |
♦ |
♦ |
♦ |
|
|
Topological space |
♦ |
◊ |
♦ |
♦ |
♦ |
|
Alexandroff space |
♦ |
◊ |
♦ |
◊ |
|
♦ |
Alexandroff topology |
♦ |
◊ |
♦ |
◊ |
♦ |
♦ |
♦: the axiom satisfied by definition ◊: the property induced by definition.
Using the dual interior operation
of
, we have the following equivalent axioms (
CL0)∼(
CL5). For
, we have
(INT0)
(INT2)
(INT3)
(INT4)
(INT5) for any family
of subsets on
,
.
where,
(INT1)+(INT3) are equivalent to
INT3′:
.
Note 7.3.1
Under the general order theory sense, the closure space is defined by axioms (CL1), (CL2) and (CL4). For example, the closure operation defined by Definition 7.1.4 is called closure operator. When considering the inclusion relation between a power set and a subset, the axiom (CL0) may be or may not be satisfied.
Note 7.3.2
Under the Cech’s sense, the closure space is called pre-topology and is defined by axioms (CL0)∼(CL3). In Definition 7.1.4, axioms (CL0) and (CL3) are replaced by (CL3)’. The topology described by the Kuratowski closure operator that satisfies axioms (CL0), (CL2), (CL3)’ and (CL4) is equivalent to the above description, since axiom (CL3)’ may induce axiom (CL3), and (CL4)+(CL3) may induce (CL3)’. The distinction between the closure space in the Cech’s sense and the topologic space in general sense is the satisfaction of the idempotent axiom or not. So the former is the extension of the latter.
Note 7.3.3
Axiom
(CL5) is called Alexandroff property. The topologic space that satisfies the Alexandroff property is called Alexandroff topology. In
Cech (1966) and Galton (2003), axiom (
CL5) is called quasi-discrete property. The Cech closure space that satisfies quasi-discrete property is called quasi-discrete closure space.
Note 7.3.4
To describe the closure space, except closure and interior operations, the neighborhood and the filter convergent sequence can be used equivalently. In
Table 7.3.1, the neighborhood and Smith spaces (
Kelly, 1955;
Smith, 1995) originally are described by neighborhood language; we use the equivalent closure axioms.