Addenda A. Some Concepts and Properties of Point Set Topology

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Addenda A

Some Concepts and Properties of Point Set Topology

A.1. Relation and Mapping

A.1.1. Relation

Definition 1.1.1

and

are any two sets.

is called a Cartesian product of X and Y, denoted by

, where

is a pair of ordered elements. x is the first coordinate of

, and y is the second coordinate of

. X is a set of the first coordinates of

, and Y is a set of the second coordinates of

Definition 1.1.2

and

are two sets. For any

, R is called a relation from X to Y.

Assume that R is a relation from X to Y. If

, then x and y are

relevant, denoted by

Set

is called the domain of R, denoted by

Set

is called the range of R, denoted by

For

, letting

is called a set of images (or image) of A.

For

, letting

is called the preimage of B.

Definition 1.1.3

For

, letting

, T is called the composition of R and S, denoted by

For

, letting

is called the inverse of R.

Proposition 1.1.1

For

and

, we have

(1)

(2)

(3)

(4)

and

(5)

Note that in (4)

rather than

A.1.2. Equivalence Relation

Definition 1.2.1

Assume that R is a relation from X to X (or a relation on X) and satisfies

(1)

(Reflexivity)

(2)

(Symmetry)

(3)

(Transitivity)

R is called an equivalence relation on X.

Assume that R is an equivalence relation on X. For

, letting

is an

equivalent set of x.

Definition 1.2.2

For

, if

and

, then

is a partition of x.

Proposition 1.2.1

R is an equivalence relation on X. Then,

is a partition of X.

A.1.3. Mapping and One–One Mapping

Definition 1.3.1

F is a relation from X to Y. For

, if there exists a unique

such that

, then F is called a mapping from X to Y, denoted by

, F is called surjective, where

is the range of F.

For

, if

, F is called 1-1 mapping.

Proposition 1.3.1

f :

is a mapping. For

, we have

, then

For

, we have

, then

If f is surjective, then

If f is a 1-1 mapping, then

Where,

is the complement of A.

is the inverse of f.

If f is surjective and 1-1 mapping, then

and

Definition 1.3.2

Assume that X is a Cartesian product of

. Let

. Define

is the projection of X on

, or a set of the i-th coordinates.

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.2. Topology Space

A.2.1. Metric Space

X is a non-empty set.

is a mapping, where R is a real set.

, d satisfies:

(1)

and

(2)

(3)

Then, d is a distance function on X and

is a metric space.

Definition 2.1.2

is a metric space. For

is called a spherical neighborhood with x as its center and

as its radius, or simply

neighborhood.

Proposition 2.1.1

is a metric space. Its spherical neighborhoods have the following properties.

(1)

, there is one neighborhood at least.

, have

(2)

, for any two spherical neighborhoods

and

, there exists

such that

(3) If

, then there exists

A.2.2. Topological Space

Definition 2.2.1

X is a non-empty set.

is a family of subsets of X. If

satisfies the following conditions

(1)

(2)

(3)

then

is a topology of X.

is a topologic space. Each member of

is called an open set on

is a metric space. For

and

, if there exists

, then A is an open set on X. Let

be a family of all open sets on X. It can be proved that

is a topology on X.

is called a topologic space induced from d.

Definition 2.2.2

is a topologic space (

always indicates a topologic space below). For

and

, if

, then U is called a neighborhood of x denoted by

For

, the set of all neighborhoods of

is called a system of neighborhoods of x, denoted by

Proposition 2.2.1

is a topologic space. For

is a neighborhood system of x. We have

(1)

and

, then

(2) If

, then

(3) If

, there exist

such that for

have

A.2.3. Induced Set, Close Set and Closure

Definition 2.3.1

For

, if

, then x is called an accumulation (limit) point of A.

Set

of all accumulation points of A is called an induced set of A.

Proposition 2.3.1

For

, we have

(1)

(2)

(3)

(4)

Definition 2.3.2

For

, if all accumulation points of A belong to A, then A is a close set.

Proposition 2.3.2

A is close

is open.

Proposition 2.3.3

Assume that

is a family of all close sets on

. We have

(1)

(2) If

, then

(3) If

₁

, then

Definition 2.3.3

For

, letting

is called a closure of A.

Proposition 2.3.4

For

, we have

(1)

(2)

(3)

(4)

Definition 2.3.4

For

, define

Proposition 2.3.5

For

, we have

(1)

(2)

A.2.4. Interior and Boundary

Definition 2.4.1

For

, letting

is called the interior (core) of A.

Proposition 2.4.1

For

, we have

(1) A is open

(2)

(3)

(4)

(5)

(6)

Definition 2.4.2

For

, if

and

, x is called a boundary point of A. The set of all boundary points of A is called boundary of A, denoted by

Proposition 2.4.2

For

, we have

(1)

(2)

(3)

(4)

(5)

A.2.5. Topological Base and Subbase

Definition 2.5.1

is a topologic space. For

and

, if there exists

such that

, then

is a base of

Proposition 2.5.1

is a space.

is a topologic space induced from d. Then,

={all spherical neighborhoods of x,

} is a base of

Proposition 2.5.2

is a family of open sets on

. Then,

is a base

and

, there is

such that

Proposition 2.5.3

is a family of subsets of X and satisfies

(1)

(2) If

, for

, there exists

such that

Then, let

be a topology of X and

be a base of

Definition 2.5.2

is a space.

is a sub-family of

. If

, letting

, i.e.,

is a family of sets composed by the intersections of any finite number of elements in

, then

is a base of

, and

is a subbase of

A.2.6. Continuous Mapping and Homeomorphism

Definition 2.6.1

is a mapping. If

, then f is a continuous mapping.

and

, have

, then f is continuous at x.

Proposition 2.6.1

For

, the following statements are equivalent.

(1) f is a continuous mapping

(2) If

is a base of Y, then

(3)

is a preimage of any close set F in Y;

is close in X.

(4)

is a subbase of Y;

, have

(5)

, have

(6)

, have

Proposition 2.6.2

For

, the following statements are equivalent.

(1) f is continuous at x.

(2) For all neighborhoods

, there exists

such that

Proposition 2.6.3

and

are continuous, then

is continuous.

Definition 2.6.3

and

are two spaces. If there exists

, where f is a 1-1 surjective and bicontinuous mapping, i.e., both f and

are continuous, then f is called a homeomorphous mapping from X to Y, or X and Y are homeomorphism.

A.2.7. Product Space and Quotient Space

Definition 2.7.1

and

are two topologies on X. If

is called smaller (coarser) than

is a family of topologies on X. If there exists

such that

, then

is called the smallest (coarsest) topology in

Similarly, we may define the concept of the largest (finest) topology.

Proposition 2.7.1

Assume that

. There exists the smallest (coarsest) topology among topologies on X that make each

continuous.

Proposition 2.7.2

Assume that

. There exists the largest (finest) topology among topologies on X that make each

continuous.

Corollary 2.7.2

Assume that

. There exists the largest (finest) topology among topologies on

that make

continuous. The topology is called the quotient topology with respect to

and

Definition 2.7.2

For

, letting

is called the subspace of

Definition 2.7.3

Assume that

, where

indicates the product set.

is a family of topologic spaces. Let

be a projection.

is the smallest topology among topologies on

that make

continuous.

is called the product topologic space of

, denoted by

Proposition 2.7.3

Assume that

is a product topologic space of

. Letting

is a subbase of

Proposition 2.7.4

Assume that

is a product topologic space of

is continuous

is continuous.

Proposition 2.7.5

Assume that

is a product topologic space of

. Then, series

on X converges to

, series

converges to

Where, the definition of convergence is that for

, if

, there exists

such that when

. Then

is called to be converging to

, denoted by

Definition 2.7.4

is an equivalence relation on

. Let

be a nature projection

, and

be the finest topology that makes

continuous.

is called the quotient space of

with respect to R. Where,

may be indicated by

, or

Proposition 2.7.6

Assume that

is a quotient topologic space of

with respect to R. Then,

Definition 2.7.5

For

, letting

is called the quotient topology of

with respect to f. We have a topologic space

and

is a congruence space of

and f.

Proposition 2.7.7

is an open (close) surjective mapping. Then,

Proposition 2.7.8

is an congruence space of

and f. Assume that

and

. Then,

is continuous

is continuous.

A.3. Separability Axiom

A.3.1. , , Spaces

Definition 3.1.1

is a space. For

, there is

such that

, or there is

such that

, X is called

space.

Definition 3.1.2

is a space. For

, there must be

such that

, X is called

space.

Definition 3.1.3

is a space. For

, there must be

such that

, X is called

space, or

space.

Proposition 3.1.1

X is a

space

, where

is the closure of singleton

. It means that the closures of any two different singletons are different.

Proposition 3.1.2

is a topologic space. The following statements are equivalent.

(1) X is a

space.

(2) Each singleton on X is a close set.

(3) Each finite set on X is a close set.

Proposition 3.1.3

is a

space

, the intersection of all neighborhoods containing x is just

Proposition 3.1.4

, X is a

space. Then,

is an infinite set.

Proposition 3.1.5

is a

space. Then, the convergent series on X has only one limit point.

Proposition 3.1.6

is a

space

the diagonal

of product topologic space on

is a close set.

3.2. , , Regular and Normal Space

Definition 3.2.1

In space

, A is close. For

, if there exist open sets

and

, such that

, then X is called a

space.

Definition 3.2.2

In space

, for

, if there exist open sets

and

such that

, then X is called a

space.

Proposition 3.2.1

is a

space

and

, there exists

such that

Proposition 3.2.2

is a

space

for any close set A in X and any open set u that contains A, i.e.,

, there exists open set

such that

Proposition 3.2.3

is a

space

For close sets

, there exists a continuous mapping

such that

and

Proposition 3.2.4 (Tietz Theorem)

is a

space

For any close set

and any continuous function

on A, there exists a continuous expansion

on X.

Definition 3.2.3

is a

and

space, then X is called a regular space.

Definition 3.2.4

is a

and

spaces, then X is called a normal space.

Proposition 3.2.5

is a normal space

is a regular space

is a

space

is a

space

is a

space.

A.4. Countability Axiom

A.4.1. The First and Second Countability Axioms

Definition 4.1.1

has countable base, then X is said to satisfy the second countability axiom.

Definition 4.1.2

If in

, for

, there exists countable local base, then X is said to satisfy the first countability axiom.

Proposition 4.1.1

Real space R satisfies the second countability axiom.

Proposition 4.1.2

is a metric space, then X satisfies the first countability axiom.

Proposition 4.1.3

satisfies the second countability axiom, then X satisfies the first countability axiom.

Proposition 4.1.4

is a continuously open and surjective mapping. If X satisfies the second (or first) countability axiom, then Y will satisfy the second (or first) countability axiom.

Definition 4.1.3

has property P and any sub-space of X also has the property P, property P is called having heredity.

If for

has property P and their product space

also has property P, then P is called having integrability.

The relation among separation axiom, countability axiom, heredity and integrability is shown in Table 4.1.1.

Table 4.1.1

								Separable	Distance
heredity	√	√	√	√	×	√	√	×	√
integrability	√	√	√	√	×	√	√	√	√(countable)

Where, and are the first and second countability axioms, respectively.
√ (countable) means that the product space of the countable number of metric spaces is metrizable.

Proposition 4.1.5

For

, if X is countable, then f is continuous at

, have

A.4.2. Separable Space

Definition 4.2.1

and

, then D is called dense in X, or D is a dense subset of X.

Proposition 4.2.1

Assume that D is a dense subset in

and

are two continuous mappings. Then,

on D.

Definition 4.2.2

has dense countable subsets, X is called a separable space.

Proposition 4.2.1

satisfies

, then X is separable.

Proposition 4.2.2

If a separable metric space satisfies

, then it must be

The relation among

and metric spaces is shown below.

Where, A

C indicates that property A with the addition of property B infers property C.

A.4.3. Lindelof Space

Definition 4.3.1

is a family of sets and B is a set. If

, then

is called a cover of set B. When

is countable or finite,

is called a countable or finite cover.

If a family

of sets covers B and sub-family

also covers B, then

is called a sub-cover of

If each set of cover

is open (closed), then

is called an open (closed) cover.

Definition 4.3.2

, for any open cover of X, there exists countable sub-cover, X is called a Lindelof space.

Proposition 4.3.1

satisfies

, then X is a Lindelof space.

Corollary 4.3.1

An n-dimensional Euclidean space

is a Lindelof space.

Proposition 4.3.2

is a Lindelof space, then X satisfies

Proposition 4.3.3

If any sub-space in

is a Lindelof space, then each uncountable subset A of X must have accumulation points of A.

A.5. Compactness

A.5.1. Compact Space

Definition 5.1.1

, if each open cover of X has its finite sub-covers, then X is called a compact space.

Definition 5.1.2

Assume that

is a family of sets. If each finite sub-family in

has non-empty intersection, then

is said to have the finite intersection property.

Proposition 5.1.1

is compact ⇔ each family of close sets that has the finite intersection property in X has non-empty intersection.

Proposition 5.1.2

is a continuous mapping. If X is compact, then

is also compact.

Proposition 5.1.3

Each close subset of a compact set is compact.

Proposition 5.1.4

, is compact, then their product space is compact as well.

A.5.2. Relation between Compactness and Separability Axiom

Proposition 5.2.1

A compact subset in

is close.

Proposition 5.2.2

A compact

space is a normal space.

Proposition 5.2.3

is a continuous mapping. If X is compact and Y is

, then f is a close mapping, i.e., mapping a close set to a close set.

Proposition 5.3.4

is a continuous and 1-1 surjective mapping. If X is compact and Y is

, then f is homeomorphous.

Proposition 5.2.5

and

is an n-dimensional Euclidean space, then A is compact

A is a bounded close set.

Proposition 5.2.6

is a continuous mapping. If X is compact, there exist

such that

A.5.3. Some Relations in Compactness

Definition 5.3.1

A topological space

is countably compact if every countable open cover has a finite subcover.

Definition 5.3.2

A topological space

is said to be limit point compact if every infinite subset has a limit point.

Definition 5.3.3

A topological space

is sequentially compact if every infinite sequence has a convergent subsequence.

In metric space, especially in n-dimensional Euclidean space, the four concepts of compactness, limit point compactness, countable compactness, and sequential compactness are equivalent.

A.5.4. Local Compact and Paracompact

Definition 5.4.1

, for each point on X there exists a compact neighborhood, and X is called a local compact space.

Definition 5.4.2

Assume that

and

are two covers of X. If each member of

is contained by some member of

, then

is called the refinement of

Definition 5.4.3

is a cover of subset A. If

, there exists

such that

only intersects with the finite number of members in

, then

is called a the local finite cover of A.

Definition 5.4.4

, for each open cover of

on X, there exists local finite cover

, where

is the refinement of

, then X is called a paracompact space.

Proposition 5.4.1

Each locally compact and

space are normal spaces.

Proposition 5.4.2

Each paracompact normal space is a regular space.

The relation among compactness, paracompactness and local compactness is shown below.

A.6. Connectedness

A.6.1. Connected Space

Definition 6.1.1

Assume that

. If

, then

and

are separate subsets.

Definition 6.1.2

, if there exist non-empty separate subsets

and

on X such that

, then X is said to be disconnected. Non-disconnected spaces are called connected spaces.

Proposition 6.1.1

, the following conditions are equivalent.

(1) X is disconnected

(2) X can be represented by the union of two non-empty and mutually disjoint close sets, i.e.,

, where

and

are non-empty close sets

(3) X can be represented by the union of two non-empty and mutually disjoint open sets.

(4) There exists non-empty both open and close proper subset on X.

Definition 6.1.3

For

, if A is regarded as a sub-space of X, then it’s connected; A is called a connected subset of X.

Proposition 6.1.2

is disconnected

there exist non-empty separate subsets

and

on X and

Proposition 6.1.3

Assume that Y is a connected subset on

. If

and

are separate subsets on X,

, then

Proposition 6.1.4

Assume that

is a connected subset. Let

. Then, B is a connected subset, especially

is connected.

Proposition 6.1.5

Assume that

is a family of connected sets on

and

. Then,

is connected.

Proposition 6.1.6

is a continuous mapping. If X is connected, then

is connected on Y.

Proposition 6.1.7

are connected spaces, then their product space

is also connected.

From

is connected,

is connected.

Proposition 6.1.8

is continuous, X is connected, and there exist

such that

, then for

, there must have

such that

Proposition 6.1.9

is a continuous mapping, where

is a unit circle, then there exists

such that

, where

A.6.2. Connected Component and Local Connectedness

Definition 6.2.1

Assume that

and

are two points on topologic space

. If there exists a connected set

such that

, then

and

are called connected.

The connected relation among points on

is an equivalence relation.

Definition 6.2.2

Each equivalent class with respect to connected relations on

is called a connected component of X.

Definition 6.2.3

For

, if A is regarded as a sub-space, its connected component is called a connected component of subset A of X.

Definition 6.2.4

, for each neighborhood u of

, there exist connected neighborhood

such that

, then X is called local connected at point x. If for

is local connected at x, then X is called a local connected space.

Proposition 6.2.1

, C is a connected component of X, then

(1) If Y is a connected subset on X and

, then

(2) C is a connected subset.

(3) C is a close set on X.

Proposition 6.2.2

, the following statements are equivalent.

(1) X is a local connected space.

(2) Any connected component of any open set of X is open.

(3) There exists a base on X such that its each member is connected.

Proposition 6.2.3

is a continuous mapping. X is local connected. Then,

is also local connected.

Proposition 6.2.4

are local connected spaces, then their product space is also local connected.

Proposition 6.2.5

is a connected open set, then A must be a connected component of

A.6.3. Arcwise Connected Space

Definition 6.3.1

is a continuous mapping and is called an arc (or path) that connects points

and

, where

and

are called start and end points of arc

, respectively. If

, then f is called a circuit.

If f is an arc on X, then

is called a curve on X.

For

, if there exists an arc

such that

and

, then X is an arcwise connected space.

For

, regarding A as a sub-space, if A is arcwise connected, then A is an arcwise connected subset of X.

Definition 6.3.2

For

, if there is an arc on X that connects

and

, then

and

are arcwise connected.

All points on X are an equivalent relation with respect to arcwise connected relations.

Definition 6.3.3

The points on

that belong to an equivalent class with respect to arcwise connected relations are called an arcwise connected component of X.

Proposition 6.3.1

is arcwise connected, then X is connected.

Proposition 6.3.2

is a continuous mapping. If X is an arcwise connected space, then

is also an arcwise connected space.

Proposition 6.3.3

are arcwise connected spaces, then their product space is also an arcwise connected space.

Corollary 6.3.3

is an arcwise connected space.

Proposition 6.3.4 (Bond Lemma)

Assume that

are close sets and

and

are continuous mappings.

, i.e.,

and

are the same on

. Let

. Then,

is continuous.

Proposition 6.3.5

For

, if A is an open connected set, then A is arcwise connected.

Definition 6.3.4

for any neighborhood

of x, if there exists a connected neighborhood

such that

, then X is called local arcwise connected.

Proposition 6.3.6

is local arcwise connected and A is connected, then A is arcwise connected.

Proposition 6.3.7

is local arcwise connected and

is an open connected subset, then A is arcwise connected.

Proposition 6.3.8

The continuous image of a local arcwise connected space is also local arcwise connected.

Proposition 6.3.9

are local arcwise connected, then their product space is also local arcwise connected.

Definition 6.3.5

is a metric space. For

and

, if there exist a set of points

such that

, then

is called a

chain that connects points

and

The above materials are from [Xio81]. The interested readers can also refer to [Eis74].

A.7. Order-Relation, Galois Connected and Closure Space

A.7.1. Order-Relation and Galois Connected

Definition 7.1.1

Assume that ‘

’ is a binary relation on

and satisfies reflexivity and transitivity properties, i.e.,

and

, if

and

, then

, ‘

’ is called a pre-order or quasi-order on

Especially, if ‘

’ satisfies transitivity and anti-reflexivity, i.e., for

does not hold, then ‘

’ is a strict pre-order on

denoted by ‘

’ generally.

Definition 7.1.2

If a pre-order relation satisfies anti-symmetry, i.e.,

, then

is called a partial order relation on

is called a partial ordered set.

If a pre-order relation

satisfies symmetry, i.e.,

, then

is called an equivalence relation on

. Symbol

is not used to denote equivalence relations generally.

Definition 7.1.3

Assume that

is a semi-order (partial-order) relation on

. For any two elements

, if their supremum

and infimum

exist, then

is a lattice. For a lattice

and

are used to represent the supremum and infimum of two elements

and

generally.

Especially, if for any

and

exist, then

is called a complete lattice.

Definition 7.1.4

is a semi-order set.

, if self-mapping

satisfies the following conditions

(1)

(increasing property)

(2)

(order-preserving)

(3)

(idempotent)

Then,

is a closure operator on

. Correspondingly, if a self-mapping

satisfies order-preserving, idempotent and decreasing property, i.e.,

, then

is called an interior operator on

Note 7.1.1

is a closure operator on

. A set

of images is just a set composed by all fixed points of

, i.e.,

. Elements of

are called to be closed under the mapping

. Especially, if

is a complete lattice, then

is also a complete lattice.

Note 7.1.2

Assume that

is a closure operator on a complete lattice

, where

is any given set and

is a power set of

. Then,

uniquely corresponds to a family

of subsets of

and

satisfies (1)

, (2)

is called a Moore family on

, and two-tuple

is a closure system.

Please refer to Davey and Priestley (1992) for more details.

Galois Connection (Davey and Priestley, 1992)

Definition 7.1.5

Assume that

and

are a pair of semi-order structures.

and

are a pair of mappings. The domains of

and

are

and

, respectively. If

and

satisfy

For

and

Then,

is called a Galois connection between

and

as shown below,

Proposition 7.1.1

is a Galois connection between

and

, where

and

. If

and

, then we have the following conclusions.

(1)

(2)

(3)

Conversely, assume that

and

are a pair of mappings between

and

. For

and

, the above two conditions (1) and (2) hold. Then,

and

are a Galois connection between

and

Proposition 7.1.2

Assume that

is a Galois connection between

and

, where

and

. Then the combination mapping

is a closure operator on

, and

is an interior operator on

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)

(cl3)

and

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

is a closure space, then

(1)

(2) For

and

, if

, then

(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

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

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)

(int3)

and

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

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

is a topological closure operation, then closure space

is a topological space.

Proposition 7.2.5

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

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

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

, or

is a homeomorph of

Definition 7.2.7

If there exists a homeomorphous mapping from closure space

, 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

such that

, and if

, 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)

(CL1)

(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

, we have the following equivalent axioms (CL0)∼(CL5). For

, we have

(INT0)

(INT1)

(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.

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Table of Contents for Addenda A. Some Concepts and Properties of Point Set Topology

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A.1. Relation and Mapping

A.1.1. Relation

Definition 1.1.1

Definition 1.1.2

Definition 1.1.3

Proposition 1.1.1

A.1.2. Equivalence Relation

Definition 1.2.1

Definition 1.2.2

Proposition 1.2.1

A.1.3. Mapping and One–One Mapping

Definition 1.3.1

Proposition 1.3.1

Definition 1.3.2

A.1.4. Finite Set, Countable Set and Uncountable Set

Definition 1.4.1

Theorem 1.4.1 (Bernstein)

A.2. Topology Space

A.2.1. Metric Space

Definition 2.1.2

Proposition 2.1.1

A.2.2. Topological Space

Definition 2.2.1

Definition 2.2.2

Proposition 2.2.1

A.2.3. Induced Set, Close Set and Closure

Definition 2.3.1

Proposition 2.3.1

Definition 2.3.2

Proposition 2.3.2

Proposition 2.3.3

Definition 2.3.3

Proposition 2.3.4

Definition 2.3.4

Proposition 2.3.5

A.2.4. Interior and Boundary

Definition 2.4.1

Proposition 2.4.1

Definition 2.4.2

Proposition 2.4.2

A.2.5. Topological Base and Subbase

Definition 2.5.1

Proposition 2.5.1

Proposition 2.5.2

Proposition 2.5.3

Definition 2.5.2

A.2.6. Continuous Mapping and Homeomorphism

Definition 2.6.1

Proposition 2.6.1

Proposition 2.6.2

Proposition 2.6.3

Definition 2.6.3

A.2.7. Product Space and Quotient Space

Definition 2.7.1

Proposition 2.7.1

Proposition 2.7.2

Corollary 2.7.2

Definition 2.7.2

Definition 2.7.3

Proposition 2.7.3

Proposition 2.7.4

Proposition 2.7.5

Definition 2.7.4

Proposition 2.7.6

Definition 2.7.5

Proposition 2.7.7

Proposition 2.7.8

A.3. Separability Axiom

A.3.1. , , Spaces

Definition 3.1.1

Definition 3.1.2

Definition 3.1.3

Proposition 3.1.1

Proposition 3.1.2

Proposition 3.1.3

Proposition 3.1.4

Table of Contents for
Addenda A. Some Concepts and Properties of Point Set Topology