paper 04-17-2009. 89KB Jun 04 2011 12:03:06 AM

ACTA UNIVERSITATIS APULENSIS

No 17/2009

C0 -SPACES AND C1 -SPACES

Chawalit Boonpok

Abstract. The aim of this paper is to introduce the concepts of C0 -spaces
and C1 -spaces and study its basic properties in closure spaces.
2000 Mathematics Subject Classification: 54A05
Keywords and phrases: closure operator; closure space; C0 -space; C1 -space.
1. Introduction
Closure spaces were introduced by E. Čech [2] and then studied by many
authors, see e.g. [3,4,5,6]. M. Caldas and S. Jafari [1] introduced the notions of
∧δ − R0 and ∧δ − R1 topological spaces as a modification of the known notions
of R0 and R1 topological spaces. In this paper, we introduce the concepts of
C0 -spaces and C1 -spaces and study its basic properties in closure spaces.
2. Preliminaries
A map u : P (X) → P (X) defined on the power set P (X) of a set X is
called a closure operator on X and the pair (X, u) is called a closure space if

the following axioms are satisfied :
(N1) u∅ = ∅,
(N2) A ⊆ uA for every A ⊆ X,
(N3) A ⊆ B ⇒ uA ⊆ uB for all A, B ⊆ X.

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C. Boonpok - C0 -Spaces and C1 -Spaces
A closure operator u on a set X is called additive (respectively,idempotent)
if A, B ⊆ X ⇒ u(A ∪ B) = uA ∪ uB ( respectively, A ⊆ X ⇒ uuA = uA). A
subset A ⊆ X is closed in the closure space (X, u) if uA = A and it is open if
its complement is closed. The empty set and the whole space are both open
and closed.
A closure space (Y, v) is said to be a subspace of (X, u) if Y ⊆ X and
vA = uA ∩ Y for each subset A ⊆ Y . If Y is closed in (X, u), then the
subspace (Y, v) of (X, u) is said to be closed too.
Let (X, u) and (Y, v) be closure spaces. A map f : (X, u) → (Y, v) is said
to be continuous if f (uA) ⊆ vf (A) for every subset A ⊆ X.
One can see that a map f : (X, u) → (Y, v) is continuous if and only if
−1

uf (B) ⊆ f −1 (vB) for every subset B ⊆ Y . Clearly, if f : (X, u) → (Y, v) is
continuous, then f −1 (F ) is a closed subset of (X, u) for every closed subset F
of (Y, v).
Let (X, u) and (Y, v) be closure spaces and let f : (X, u) → (Y, v) be a
map. If f is continuous, then f −1 (G) is an open subset of (X, u) for every
open subset G of (Y, v).
Let (X, u) and (Y, v) be closure spaces. A map f : (X, u) → (Y, v) is said
to be closed ( resp. open ) if f (F ) is a closed (resp. open) subset of (Y, v)
whenever F is a closed (resp. open) subset of (X, u).
uα ) : α
∈ I} of Q
closure spaces, denoted by
Q The product of a family {(Xα ,Q
Xα denotes the cartesian
Xα , u where
(Xα , uα ), is the closure space
α∈I

α∈I

α∈I

product of sets XQ
α , α ∈ I, and u is the closure operator generated by the
(Xα , uα ) → (Xα , uα ), α ∈ I, i.e., is defined by uA =
projections πα :
α∈I
Q
Q
uα πα (A) for each A ⊆
Xα .
α∈I

α∈I

The following statement is evident:
Proposition 1. Let {(Xα , uα ) : α ∈ I}Qbe a family of closure spaces and let
(Xα , uα ) → (Xβ , uβ ) is closed and
β ∈ I. Then the projection map πβ :
α∈I

continuous.
Proposition 2. Let {(Xα , uα ) : α ∈ I} be a family of closure spaces
Q and let
Xα is a
β ∈ I . Then F is a closed subset of (Xβ , uβ ) if and only if F ×
α6=β
α∈I

closed subset of

Q

(Xα , uα ).

α∈I

38

C. Boonpok - C0 -Spaces and C1 -Spaces
Proof. Let β ∈ I and let F be a closed

is conβ ). Since πβ Q
Q subset of (Xβ , u−1
Xα ,
(Xα , uα ). But πβ (F ) = F ×
tinuous, πβ−1 (F ) is a closed subset of
α∈I

hence F ×

Q

α6=β
α∈I

Q

Xα is a closed subset of

(Xα , uα ).

α∈I

α6=β
α∈I

Conversely, let F ×

Q

Q

Xα be a closed subset of

(Xα , uα ). Since πβ is

α∈I

α6=β
α∈I



Q
Xα = F is a closed subset of (Xβ , uβ ).
closed, πβ F ×
α6=β
α∈I

Proposition 3. Let {(Xα , uα ) : α ∈ I} be a family of closure spaces
Q and let
Xα is an
β ∈ I. Then G is an open subset of (Xβ , uβ ) if and only if G ×
α6=β
α∈I

Q

open subset of

(Xα , uα ).

α∈I

Proof. Let β ∈ I and let G be an open
is conQ subset of (Xβ , u−1β ). Since πβ Q
−1
tinuous, πβ (G) is an open subset of
Xα ,
(Xα , uα ). But πβ (G) = G ×
α∈I

therefore G ×

Q

Xα is an open subset of

Conversely, let G×

Q

Xα be an open subset of

Xα is a closed subset of

(Xβ − G) ×

Q

(Xα , uα ). But

α∈I

α6=β
α∈I

Q

α6=β
α∈I

Q

(Xα , uα ). Then

Xα , hence (Xβ − G) ×

Q

Xα − G ×

α∈I

Q

Q

Xα −

α∈I

α∈I

α6=β
α∈I

Q

(Xα , uα ).

α∈I

α6=β
α∈I

G×

α6=β
α∈I

Q

Xα is a closed subset of

α6=β
α∈I

Q

Xα =

α6=β
α∈I

Q

(Xα , uα ).

α∈I

By Proposition 2, Xβ − G is a closed subset of (Xβ , uβ ). Consequently, G is
an open subset of (Xβ , uβ ).

3. C0 -Spaces and C1 -Spaces
Definition 4. A closure space (X, u) is said to be a C0 -space if, for every open
subset G of (X, u), x ∈ G implies u{x} ⊆ G.
Proposition 5. A closure space (X, u) is a C0 - space if and only if, for every
closed subset F of (X, u) such that x ∈
/ F , u{x} ∩ F = ∅.

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C. Boonpok - C0 -Spaces and C1 -Spaces
Proof. Let F be a closed subset of (X, u) such that x ∈
/ F . Then X − F is
an open subset of (X, u) such that x ∈ X − F . Since (X, u) is a C0 -space,
u{x} ⊆ X − F . Consequently, u{x} ∩ F = ∅.
Conversely, let G be an open subset of (X, u) and let x ∈ G. Then X −G is
a closed subset of (X, u) such that x ∈
/ X − G . Therefore, u{x} ∩ (X − G) = ∅.
Consequently, u{x} ⊆ G. Hence, (X, u) is a C0 -space.
Definition 6. A closure space (X, u) is said to be a C1 -space if, for each
x, y ∈ X such that u{x} =
6 u{y}, there exists disjoint open subsets U and V
of (X, u) such that u{x} ⊆ U and u{y} ⊆ V .
Proposition 7. Let (X, u) be a closure space. If (X, u) is a C1 -space, then
(X, u) is a C0 -space.
Proof. Let U be an open subset of (X, u) and let x ∈ U . If y ∈
/ U , then
u{x} =
6 u{y} because x ∈
/ u{y}. Then there exists an open subset Vy of (X, u)
such that u{y} ⊆ Vy and x ∈
/ Vy , which implies y ∈
/ u{x}. Thus, u{x} ⊆ U .
Hence, (X, u) is a C0 -space.
The converse is not true as can be seen from the following example.
Example 8. Let X = {a, b, c} and define a closure operator u : P (X) → P (X)
on X by u∅ = ∅, u{a} = {a}, u{b} = u{c} = {b, c} and u{a, b} = u{a, c} =
u{b, c} = uX = X. Then (X, u) is a C0 -space but not a C1 -space.
Proposition 9. A closure space (X, u) is a C1 -space if and only if, for every
pair of points x, y of X such that u{x} 6= u{y}, there exists open subsets U
and V of (X, u) such that x ∈ U , y ∈ V and U ∩ V = ∅.
Proof. Suppose that (X, u) is a C1 -space. Let x, y be points of X such that
u{x} =
6 u{y}. There exists open subsets U and V of (X, u) such that x ∈
u{x} ⊆ U and y ∈ u{y} ⊆ V .
Conversely, suppose that there exists open subsets U and V of (X, u) such
that x ∈ U , y ∈ V and U ∩ V = ∅. Since every C1 -space is C0 -space, u{x} ⊆ U
and u{y} ⊆ V . This gives the statement.
Proposition
10. Let {(Xα , uα ) : α ∈ I} be a family of closure spaces. If
Q
(Xα , uα ) is a C0 -space, then (Xα , uα ) is a C0 -space for each α ∈ I.
α∈I

40

C. Boonpok - C0 -Spaces and C1 -Spaces
Q

(Xα , uα ) is a C0 -space. Let β ∈ I and let G be an
Q
Xα is an open subset
open subset of (Xβ , uβ ) such that xβ ∈ G. Then G ×

Proof. Suppose that

α∈I

α6=β
α∈I

of

Q

(Xα , uα ) such that (xα )α∈I ∈ G ×

α∈I

space,

Q

Xα . Since

α6=β
α∈I

Q

uα πα ({(xα )α∈I }) ⊆ G ×

α∈I

Q

Q

(Xα , uα ) is a C0 -

α∈I

Xα . Consequently, uβ {xβ } ⊆ G. Hence,

α6=β
α∈I

(Xβ , uβ ) is a C0 -space.
Proposition 11. Let {(Xα , uα ) : α ∈ I} be
Q a family of closure spaces. If
(Xα , uα ) is a C1 -space.
(Xα , uα ) is a C1 -space for each α ∈ I, then
α∈I

Proof. Suppose thatQ(Xα , uα ) is a CQ
1 -space for each α ∈ I.QLet (xα )α∈I and
(yα )α∈I be points of
Xα such that
uα πα ({(xα )α∈I }) 6=
uα πα ({(yα )α∈I }).
α∈I

α∈I

α∈I

There exist β ∈ I such that uβ {xβ } 6= uβ {yβ }. Since (Xβ , uβ ) is a C1 -space,
there exists open subsets U and V of (Xβ , uβ ) such that uβ {xβ } ⊆ U and
uβ {yβ } ⊆ V . Consequently,
Q
Q
Q
Q
Xα
Xα ,
uα πα ({(yα )α∈I }) ⊆ V ×
uα πα ({(xα )α∈I }) ⊆ U ×
α∈I

such that U ×

Q

Xα and V ×

α6=β
α∈I

Q

α∈I

α6=β
α∈I

Q

Xα are open subsets of

α6=β
α∈I

Q

(Xα , uα ). Hence,

α∈I

α6=β
α∈I

(Xα , uα ) is a C1 -space.

α∈I

References
[1] M. Caldas and S. Jafari, On some low separation axioms in topological
spaces, Hoston Journal of Math., 29, (2003), 93-104.
[2] E. Čech,Topological Spaces, Topological Papers of Eduard Čech, Academia,
Prague (1968), 436-472.
[3] J. Chvalina, On homeomorphic topologies and equivalent set-systems,
Arch. Math. 2, Scripta Fac. Sci. Nat. UJEP Brunensis, XII, (1976), 107-116.
[4] J. Chvalina, Stackbases in power sets of neighbourhood spaces preserving
the continuity of mappings, Arch. Math. 2, Scripta Fac. Sci. Nat. UJEP
Brunensis, XVII, (1981), 81-86.
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C. Boonpok - C0 -Spaces and C1 -Spaces
[5] L. Skula, Systeme von stetigen abbildungen, Czech. Math. J., 17, (92),
(1967), 45-52.
[6] J. Šlapal, Closure operations for digital topology, Theoret. Comput.
Sci., 305, (2003), 457-471.
Author:
Chawalit Boonpok
Department of Mathematics
Faculty of Science
Mahasarakham University
Mahasarakham 44150
Thailand
email:chawalit boonpok@hotmail.com

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