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East-West J. of Mathematics: Vol. 3, No 2 (2001) pp. 171-177
A SEPARATION AXIOM BETWEEN PRE-T0
AND PRE-T1
Miguel Caldas
Departamento de Matemática Aplicada
Universidade Federal Fluminense
Rua Mário Santos Braga, s/n CEP:24020-140, Niteroi-RJ Brasil
e-mail: [email protected]ff.br
Abstract
The author introduces a new separation axiom pre-D1 and studies
some of its basic properties. The implications of this new separation
axiom with the well-known axioms called pre-T2 , pre-T1 and pre-T0 are
obtained .
1
Introduction
Pre–open sets were introduced and investigated by A. S. Mashhour, et al [8] in
1982. In 1990, A. Kar and P. Bhattacharyya [6] used pre-open sets to define
and investigate three new separation axioms, called pre-T2 , pre-T1 and pre-T0 .
Moreover, they have shown that the following implications hold.
T2
↓
T1
↓
T0
−→
−→
−→
pre-T2
↓
pre-T1
↓
pre-T0
Later, in 1996 H. Maki, et al [7] used pre-open sets to define the axiom preT1/2 and proved that every topological space is pre-T1/2 and further investigated
the separation axioms pre-T2 , pre-T1 and pre-T0 . For other properties of this
and other weak separation axioms see by example ([1], [2] , [3], [5], [9]). The
purpose of this paper is to introduce a new separation axiom pre-D1 which is
Key words and phrases: pre-T0 space, pre-T1 space, pre-T2-space, pre-D1,pre-irresolute
mapping, pre-open set, .
(1991) Mathematics Subject Classification: 54D10.
171
172
A Separation Axiom between pre-T0 and pre-T1
strictly between pre-T0 and pre-T1 , and discuss its relations with the axioms
mentioned above.
Listed below are definitions that will be utilized. We identify the separation
axioms with the class of topological spaces satisfying these axioms.
Definition 1 Let A be a subset of a topological space (X, τ ). Then
(1) A is pre-open [8] if A ⊂ Int(Cl(A)). The family of all pre-open set will
denoted by P O(X, τ ).
(2) A is pre-closed [8] if its complement Ac (or X\A) is pre-open or equivalently if Cl(Int(A)) ⊂ A. The pre-closure of a subset B of X [4], denoted by
pCl(B), is the intersection of all pre-closed sets containing B.
(3) A is pg-closed [7] if pCl(A) ⊂ P holds whenever A ⊂ P and P ∈
P O(X, τ ).
Definition 2 A topological space (X, τ ) is pre-T0 (resp. pre-T1 ) [6] if for x, y ∈
X such that x = y, there exists a pre-open set containing x but not y or (resp.,
and) a pre-open set containing y but not x.
Definition 3 A topological space (X, τ ) is pre-T2 [6] if for x, y ∈ X such that
x = y, there exist pre-open sets P1 and P2 such that x ∈ P1 , y ∈ P2 and
P1 ∩ P2 = ∅.
2
The separation axiom pre-D1
Definition 4 Let X be a topological space. A subset S ⊂ X is called a preDifference set (in short pD-set) if there are two pre-open sets P1 , P2 in X
such that P1 = X and S = P1 \ P2 .
A pre-open set P = X is pD-set since P = P \ ∅.
If we replace pre-open sets in the usual definitions of pre-T0 , pre-T1 , pre-T2
with pD-sets, we obtain separation axioms pre-D0 , pre-D1 , pre-D2 respectively as follows.
Definition 5 A topological space (X, τ ) is pre-D0 (resp. pre-D1 ) if for x, y ∈
X such that x = y there exists an pD-set of X containing x but not y or (resp.,
and) a pD-set containing y but not x.
Definition 6 A topological space (X, τ ) is pre-D2 if for x, y ∈ X such that
x = y there exist disjoint pD-sets S1 and S2 such that x ∈ S1 and y ∈ S2 .
Remark 2.1 (i) If (X, τ ) is pre-Ti , then (X, τ ) is pre-Di , i = 0, 1, 2.
(ii) If (X, τ ) is pre-Di , then (X, τ ) is pre-Di−1 , i = 1, 2.
Miguel Caldas
173
Theorem 2.2 Let (X, τ ) be a topological space. Then,
(i) (X, τ ) is pre-D0 iff (X, τ ) is pre-T0 .
(ii) (X, τ ) is pre-D1 iff (X, τ ) is pre-D2 .
Proof (i) (⇐): Remark 2.1(i).
(⇒): Let (X, τ ) be pre-D0 . Then for each distinct pair x, y ∈ X, at least one
of x, y, say x, belongs to a pD-set S but y ∈
/ S. Let S = P1 \ P2 where P1 = X
and P1 ,P2 ∈ P O(X, τ ). Then x ∈ P1 , and for y ∈
/ S we have two cases: (a)
y∈
/ P1 ; (b) y ∈ P1 and y ∈ P2 .
In case (a), P1 contains x but does not contain y ;
In case (b), P2 contains y but does not contain x.
Hence X is pre-T0 .
(ii) (⇐): Remark 2.1(ii).
(⇒): Suppose X pre-D1 . Then for each distinct pair x, y ∈ X, we have pD-sets
S1 , S2 such that x ∈ S1 , y ∈
/ S1 ; y ∈ S2 , x ∈
/ S2 . Let S1 = P1 \P2 , S2 = P3 \P4 .
From x ∈
/ S2 we have either x ∈
/ P3 or x ∈ P3 and x ∈ P4 . We discuss the two
cases separately.
/ S1 we have two subcases:
(1) x ∈
/ P3 . From y ∈
(a) y ∈
/ P1 . From x ∈ P1 \P2 we have x ∈ P1 \ (P2 ∪ P3 ) and from y ∈ P3 \P4
we have y ∈ P3 \(P1 ∪P4 ) . It is easy to see that (P1 \(P2 ∪P3))∩(P3 \(P1 ∪P4 ) =
∅.
(b) y ∈ P1 and y ∈ P2 . We have x ∈ P1 \P2 , y ∈ P2 . (P1 \P2 ) ∩ P2 = ∅.
(2) x ∈ P3 and x ∈ P4 . We have y ∈ P3 \P4 , x ∈ P4 . (P3 \P4 ) ∩ P4 = ∅.
From the discussion above we know that the space X is pre-D2 .
2
The following example shows that pre-T0 space need not be pre-D1 .
Example 2.3 Let X = {a, b} with topology τ = {∅, {a}, X}. Then (X, τ ) is
pre-T0 and T0 , but not pre-D1 since there is not a pD-set containing b but not
a.
The following example shows that pre-D1 space need not be pre-T1 .
Example 2.4 Let X = {a, b, c} with topology τ = {∅, {a}, X}. Then we have
P O(X, τ ) = {∅, {a}, {a, b}, {a, c}, X}. Since {b} = {a, b}\{a}, {c} = {a, c}\{a},
hence (X, τ ) is a pre-D1 space but not pre-T1 since each pre-open set containing
b contains a.
Recall, that a subset Bx of a topological space X is said to be a preneighborhood of a point x ∈ X [6], iff there exists a pre-open set P such that
x ∈ P ⊂ Bx .
Definition 7 Let xo be a point in a topological space X. If xo has not preneighborhood other than X, then we call xo an pc.c (common to all pre-closed
sets point).
174
A Separation Axiom between pre-T0 and pre-T1
The name comes from the fact that xo belongs to each pre-closed set in X.
Theorem 2.5 A pre-T0 space X is pre-D1 iff there is not pc.c point in X.
Proof (⇒): If X is pre-D1 then each point x ∈ X , belongs to a pD -set
S = P1 \P2 , hence x ∈ P1 . Since P1 = X, x is not an pc.c point.
(⇐): If X is pre-T0 , then for each distinct pair of points x, y ∈ X, at least
one of x, y, say x has a pre-neighborhood U such that x ∈ U and y ∈
/ U, hence
U = X is a pD-set. If X has not a pc.c point , then y is not a pc.c point,
hence there exists a pre-neighborhood V of y such that V = X. Now y ∈ V \U ,
x∈
/ V \U and V \U is a pD-set. Therefore X is pre-D1 .
2
Corollary 2.6 A pre-T0 space X is not pre-D1 iff there is an unique pc.c point
in X.
Proof Only the uniqueness of the pc.c point is to be proved. If xo and yo are
two pc.c points in X then since X is pre-T0 , at least one of xo and yo say xo ,
has a pre-neighborhood U such that xo ∈ U, yo ∈
/ U, hence U = X, xo is not a
pc.c point, contradiction.
2
Remark 2.7 From ([6], Examples 1, 2, 5 and 6), Example 2.3 and Example
2.4, we have the following diagram:
T0
↓
↓
↑
T1
↑
T2
→
←
→
←
→
←
pre-T0
↓
↓
↑
pre-T1
↑
pre-T2
→
←
pre-D1
where A → B (resp. A → B) represents that A implies B (resp. A does not
always imply B).
Recall that, a topological space (X, τ ) is D1 [10] if for x, y ∈ X such that
x = y there exists a D-set of X containing x but not y and a D-set containing
y but not x , where a subset S ⊂ X is called a D-set if there are two open sets
O1 , O2 in X such that O1 = X and S = O1 \ O2 .
The following example shows that pre-D1 space does not imply D1 space.
Example 2.8 Let X = {a, b} with the indiscrete topology τ . Then X is pre-D1
space but not D1 .
Miguel Caldas
175
Definition 8 A topological space (X, τ ) will be termed pre-symmetric if for x
and y in X, x ∈ pCl({y}) implies y ∈ pCl({x}).
Theorem 2.9 Let (X, τ ) be a topological space. Then,
(i) (X, τ ) is pre-symmetric iff {x} is pg-closed for each x in X.
(ii) If (X, τ ) is a pre-T1 space, then (X, τ ) is pre-symmetric.
(iii) (X, τ ) is pre-symmetric and pre-T0 iff (X, τ ) is pre-T1 .
Proof (i) (⇐): Suppose x ∈ pCl({y}), but y ∈
/ pCl({x}). Then {y} ⊂
[pCl({x})]c and thus pCl({y}) ⊂ [pCl({x})]c. Then x ∈ [pCl({x})]c , a contradiction.
(⇒): Suppose that {x} ⊂ P and P is pre-open in (X, τ ) but pCl({x}) is not
a subset of P . Then pCl({x}) ∩ P c = ∅; take y ∈ pCl({x}) ∩ P c . Therefore
x ∈ pCl({y}) ⊂ P c and x ∈
/ P , a contradiction.
(ii) In a pre-T1 space, singleton sets are pre-closed ([6], Theorem 4) and
therefore pg-closed. By (i) the space is pre-symmetric.
(iii) By (ii) it suffices to prove only the necessity. Let, then x = y and by
pre-T0 , we may assume that x ∈ P1 ⊂ {y}c for some P1 ∈ P O(X, τ ). Then
x∈
/ pCl({y}) and hence y ∈
/ pCl({x}). There exists a P2 ∈ P O(X, τ ) such that
y ∈ P2 ⊂ {x}c and (X, τ ) is a pre-T1 space.
2
Theorem 2.10 Let (X, τ ) be a pre-symmetric space. Then the following are
equivalent.
(i) (X, τ ) is pre-T0 ,
(ii) (X, τ ) is pre-D1 ,
(iii) (X, τ ) is pre-T1 .
Proof This is immediate from the previous results.
2
Definition 9 A mapping f : (X, τ ) → (Y, σ) is said to be pre-irresolute [8] if
for every pre-open set P of (Y ,σ) its inverse image f −1 (P ) is also pre-open
in (X, τ ).
Theorem 2.11 If f : X → Y is a pre-irresolute surjective mapping and S is
a pD-set in Y , then f −1 (S) is a pD-set in X.
Proof Let S be a pD-set in Y . Then there are pre-open sets P1 and P2 in
Y such that S = P1 \P2 and P1 = Y . By the pre- irresolutenes of f, f −1 (P1 )
and f −1 (P2 ) are pre-open in X. Since P1 = Y , we have f −1 (P1 ) = X. Hence
f −1 (S) = f −1 (P1 )\f −1 (P2 ) is a pD-set.
2
Theorem 2.12 If (Y, σ) is pre-D1 and f : (X, τ ) → (Y, σ) is pre-irresolute
and bijective, then (X, τ ) is pre-D1 .
176
A Separation Axiom between pre-T0 and pre-T1
Proof Suppose that Y is a pre-D1 space. Let x and y be any pair of distinct
points in X. Since f is injective and Y is pre-D1 , there exist pD-sets Sx and Sy
of Y containing f(x) and f(y) respectively, such that f(y) ∈
/ Sx and f(x) ∈
/ Sy .
By Theorem 2.11 , f −1 (Sx ) and f −1 (Sy ) are pD-sets in X containing x and y
respectively. This implies that X is a pre-D1 space.
2
We finish our article giving another characterization of pre-D1 spaces.
Theorem 2.13 A space X is pre-D1 iff for each pair of distinct points x and
y in X, there exists a pre-irresolute surjective mapping f of X onto a pre-D1
space Y such that f(x) = f(y).
Proof (⇒): For every pair of distinct points of X, it suffices to take the identity
mapping on X.
(⇐): Let x and y be any pair of distinct points in X. By hypothesis, there
exists a pre-irresolute, surjective mapping f of a space X onto a pre-D1 space
Y such that f(x) = f(y). Therefore, there exist disjoint pD-sets Sx and Sy in
Y such that f(x) ∈ Sx and f(y) ∈ Sy . Since f is pre-irresolute and surjective,
by Theorem 2.11, f −1 (Sx ) and f −1 (Sy ) are disjoint pD-sets in X containing x
and y, respectively. Hence by Theorem 2.2(ii), X is pre-D1 space.
2
Corollary 2.14 Let {Xα /α ∈ I} be any family of topological spaces. If Xα is
pre-D1 for each α ∈ I, then the product space ΠXα is pre-D1 .
Proof Let (xα ) and (yα ) be any pair of distinct points in ΠX α . Then there exists an index β ∈ I such that xβ = yβ . The natural projection Pβ : ΠXα → Xβ
is continuous and open (therefore pre-irresolute) and Pβ ((xα )) = Pβ ((yα )).
Since Xβ is pre-D1 , ΠXα is pre-D1 .
2
Acknowledgement. The author thank the referee for his help in improving
the quality of this paper.
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Miguel Caldas
177
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