Articulo 8.dvi


CUBO A Mathematical Journal
Vol.12, No¯ 02, (123–126). June 2010

A note on generalized topological spaces and preorder

Saeid Jafari

College of Vestsjaelland South,

Herrestraede 11,

4200 Slagelse, Denmark,

email: jafari@stofanet.dk

Raja Mohammad Latif

Department of Mathematics and Statistics,

King Fahd University of Petroleum and Minerals,

Dhahran 31261, Saudi Arabia.

email: raja@kfupm.edu.sa

and

Seithuti P. Moshokoa

Department of Mathematical Sciences,

University of South Africa,

P.O. Box 392, Pretoria 0003, South Africa.

email: moshosp@unisa.ac.za

ABSTRACT

This paper deals with the notion of continuity in ordered generalized topological spaces.

We also characterize general topological spaces satisfying the Hausdorff property via the

graph of the preorder assigned to the general topological space.

RESUMEN

Este art́ıculo trata de continuidad en espacios topológicos ordenados. También caracter-

izamos espacios topológicos generalizados satisfaciendo la propiedad de Hausdorff v́ıa el

grafo designado de un espacio topológico general preordenado.



124 S. Jafari. R. M. Latif and S. P. Moshokoa CUBO
12, 2 (2010)

Key words and phrases: Topological Spaces, Hausdorff, preorder.

AMS Subject Classification (2000): 54D10

1 Introduction

By a generalized topology ( briefly GT) on X [1], we mean a subset µ of the power set exp X, of X

such that ∅ ∈ µ and arbitrary union of elements of µ belongs to µ. The elements of µ are said to be

µ-open, their compliments are said to be µ-closed. Clearly, each topology is a GT, but not conversely.

As a generalization of the concept of a Hausdorff topology, we shall say that a GT, written (X, µ) is

Hausdorff if for distinct points x and y in X there exists disjoint µ-open sets U and V containing x

and y, respectively. By (X, µ, ≤) we mean a GT together with a preorder. Let A be a subset of X,

let i(A) = ∪x∈Ax
↑, where x↑ = {y ∈ X : x ≤ y}. Let d(A) = ∪x∈Ax

↓, where x↓ = {y ∈ X : y ≤ x}.

Clearly, we have A ⊆ i(A) and A ⊆ d(A) for all A ⊆ X. We say that a subset A of X is increasing

(decreasing) if i(A) = A(A = d(A)). If the preorder is discrete, then (X, µ, ≤) is just a GT, as in the

literature, see for example [1] [2] . We shall simply write OGT, for a ordered generalized topological

space (X, µ, ≤). Now let G = {(x, y) : x ≤ y}. Then G is a subset of X × X, we shall refer to G

as the graph of the preorder. We shall say that a subset G is closed if X × X − G is open in the

product generalized topology, denoted by µ × µ. That is, a subset B of X × X is (µ × µ)-open if for

every (x, y) ∈ B there exist µ-open sets U and V, such that (x, y) ∈ U × V ⊂ B, where U and V

contain x and y, respectively. We shall say that the preorder on X is (µ × µ)-closed if its graph G is

(µ × µ)-closed in X × X.

2 Results

Let A be a subset of X. We denote the µ-closure of A by cµA, and recall:

Lemma 2.1. [2] If µ is a GT on X, A ⊂ X, x ∈ X, then x ∈ cµA if and only if x ∈ U ∈ µ implies

U ∩ A 6= ∅.

Proof. Define B = {x ∈ X : x ∈ U ∈ µ ⇒ U ∩ A 6= ∅}. Clearly A ⊂ B. Further B is µ-closed,

that is X − B ∈ µ because x ∈ X − B if and only if there is Ux ∈ µ such that x ∈ Ux and Ux ∩ A 6= ∅

and then y ∈ Ux implies that y ∈ X − B so that X − B = ∪x∈X−BUx ∈ µ. Finally, if A ⊂ Q and Q is

µ-closed, then X − Q ∈ µ and (X − Q) ∩ A = ∅ so that X − Q ⊂ X − B, B ⊂ Q. Therefore B is the

smallest µ-closed set containing A, i.e. B = cµA.

Clearly, for a subset A of (X, µ) we have A ⊂ cµA. Also, cµA is µ-closed. To see this just note that

the µ-closure of A is the intersection of all µ-closed supersets of A in X. Therefore, the complement

of the µ-closure of A is µ-open, being an arbitrary union of members of µ.

Example 2.2. Let X = {1, 2, 3} and define µ = {∅, {1, 2}, {2, 3}, {1, 2, 3}}. Then (X, µ, ≤) is an

OGT , where the preorder is induced from N but it is not an ordered topological space. Also, for the

subset A = {3} we have cµA = A.

We shall need the following:



CUBO
12, 2 (2010)

A note on generalized topological spaces and preorder 125

Theorem 2.3. Let µ be a GT on X, if A ⊆ B in X, then cµA ⊆ cµB.

Proof. Let (X, µ) be a GT, and A and B be subsets of X such that A ⊆ B. If x ∈ cµA, then by

Lemma 2.1, there exists Ux ∈ µ containing x such that Ux ∩ A 6= ∅. Since A ⊆ B, we have Ux ∩ B 6= ∅.

Therefore x ∈ cµB. This shows that cµA ⊆ cµB.

We will say that (X, µ, ≤) is µ-continuous (µ-anticontinuous) is and only if for each µ-open

subset G (µ-closed subset F )i(G) and d(G) are µ-open (i(F ) and d(F ) are µ-closed). Also, we say

that (X, µ, ≤) is bicontinuous if it is both continuous and anti-continuous.

Example 2.4. Let the OGT (X, µ, ≤) be that of Example 2.1. Then for every µ-open set A and

µ-closed set B in X, the sets: i(A) and d(A) are µ-open and the sets i(B) and d(B) are µ-closed. It

follows that the OGT (X, µ, ≤) is bicontinuous.

Theorem 2.5. The space (X, µ, ≤) is anti-continuous if and only if

(i) cµ(i(A)) ⊆ i(cµ(A)) and cµd(A) ⊆ d(cµA) for every A ⊆ X.

(ii) (X, µ, ≤) is continuous if and only if i(cµA) ⊆ cµ(i(A)) and d(cµA) ⊆ cµd(A) for every

A ⊆ X.

(iii) (X, µ, ≤) is bicontinuous if and only if i(cµA) = cµ(i(A)) and d(cµA) = cµd(A) for every

A ⊆ X.

Proof. (i) Suppose that (X, µ, ≤) is anticontinuous and A be a subset of X. Note that A ⊆ cµA,

and therefore i(A) ⊆ i(cµA). We apply Theorem 2.3, to obtain cµ(i(A)) ⊆ cµ(i(cµA)). Note that

i(cµA) is µ-closed as (X, µ, ≤) is anticontinuous, we then have cµ(i(cµA)) = i(cµA). Hence cµ(i(A)) ⊆

i(cµ(A)). Dually, we can show cµd(A) ⊆ d(cµA). Conversely, suppose that cµ(i(A)) ⊆ i(cµ(A)) holds

for all A ⊆ X. Let F be a µ-closed subset of X. First, we note that i(F ) ⊆ cµi(F ) this holds even

when F is not necessarily µ-closed. On the other hand, let F be µ-closed and cµ(i(F )) ⊆ i(cµ(F ))

holds. We have cµi(F ) ⊆ i(F ). Hence cµi(F ) = i(F ). This shows that i(F ) is µ-closed. Dually, d(F )

is µ-closed.

(ii) Let (X, µ, ≤) be continuous and A be a subset of X. Then cµi(A) is µ-closed, so is that

X − cµi(A) is µ-open. Hence, X − d(X − cµi(A)) is µ-closed. Note that X − d(X − cµi(A)) is

increasing and contains A. It follows that cµA ⊆ X − d(X − cµi(A)). Since A ⊆ d(A) for all A ⊆ X,

we have

X − clµi(A) ⊆ d(X − cµi(A)).

This implies that X − d(X − cµi(A)) ⊆ cµi(A). Therefore,

cµi(A) = X − d(X − cµi(A)).

Hence i(cµA) ⊆ cµi(A). Dually, we have d(cµA) ⊆ cµd(A) Conversely, suppose that i(cµA) ⊆ cµi(A)

and d(cµA) ⊆ cµd(A) holds for all A ⊆ X. Let G be a µ-open subset of X. Then X − G is µ-closed

and cµ(X − i(G)) ⊆ X − G, hence G ⊂ X − cµ(X − i(G)). It follows that

d(cµ(X − i(G))) ⊆µ d(X − i(G)) = cµ(X − i(G))

so that cµ(X − i(G)) is decreasing. Thus i(G) ⊂ X − cµ(X − i(G)). On the other hand, it is clear

that X − i(G) ⊆ cµ(X − i(G)) and hence

X − cµ(X − i(G)) ⊆ i(G).



126 S. Jafari. R. M. Latif and S. P. Moshokoa CUBO
12, 2 (2010)

Consequently, i(G) = X − cµ(X − i(G)) is µ-open. Dually, d(G) is µ-closed.

(iii) Suppose that (X, µ, ≤) is bicontinuous. Then (X, µ, ≤) is both anticontinuous and continu-

ous. So by (i) we have cµ(i(A)) ⊆ i(cµ(A)) and cµd(A) ⊆ d(cµA) for every A ⊆ X. Also by (ii), we have

i(cµA) ⊆ cµ(i(A)) and d(cµA) ⊆ cµd(A). Together, we get i(cµA) = cµ(i(A)) and d(cµA) = cµd(A)

for every A ⊆ X. The converse is clear from (i) and (ii).

Theorem 2.6. Let (X, µ, ≤) be an OGT. Then the preorder ≤ is closed if and only if, for every two

points a, b ∈ X such that a ≤ b is false, it is possible to determine a increasing µ-open set U that

contains a and a decreasing set µ-open set V containing b, such that U ∩ V = ∅.

Proof. Let a, b ∈ such that a ≤ b is false, that is, (a, b) /∈ G. Since G is (µ × µ)-closed, we have

that X × X − G is (µ × µ)-open and (a, b) ∈ X × X − G. Now find µ-open sets Ú and V́ containing

a and b, respectively such that Ú × V́ ∩ G = ∅. Now if x ∈ Ú and y ∈ V́ , then x ≤ y is false. We set

U = i(Ú ) and V = d(V́ ). Now, Ú ⊂ U and V́ ⊂ V. Then U is µ-open increasing and contains a, and

V is µ-open decreasing and contains b. Next, we show that U ∩ V = ∅. Suppose that U ∩ V 6= ∅, let

z ∈ U ∩ V. Since z ∈ U , there exists a point x ∈ Ú such that x ≤ z. Also, z ∈ V gives a point y ∈ V́ ,

such that z ≤ y. Hence, x ≤ y follows from the fact that the ”≤” is a preorder on X. On the other

hand x ∈ Ú and y ∈ V́ implies that x ≤ y is false. So we arrive at a contradiction. We conclude that

U ∩ V = ∅. Conversely, assume that G is not closed, then find (a, b) in the µ-closure of G, such that

(a, b) /∈ G. Then consider the sets U and V containing a and b, increasing and decreasing, respectively.

Then U ∩ V contains (a, b) and satisfy U × V ∩ G 6= ∅. Observe that U × V is (µ × µ)-open. So, find

u ∈ U and v ∈ V such that (u, v) ∈ G and u ≤ v. From u ∈ U and u ≤ v results that v ∈ U , since U

is increasing; thus V belongs to both U and V , that is U and V are not disjoint.

Theorem 2.7. Let (X, µ, ≤) be an OGT. If the preorder ≤ is closed, then for every point a ∈ X, the

sets d(a) and i(a) are µ-closed.

Proof. Suppose that the preorder is closed that is its graph G is closed. Let a ∈ X. If b ∈ X −i(a),

then a ≤ b is false. By Theorem 2.6, there exists an increasing µ-open set U containing a and a

decreasing µ-open set V containing b such that U and V are disjoint. Now i(a) ⊂ U implies that

W ∩ i(a) 6= ∅

this shows that i(a) is closed.

Acknowledgement. The second author is highly and greatly indebted to the King Fahd Uni-

versity of Petroleum and Minerals, for providing necessary research facilities during the preparation of

this paper. The third author has been supported by the South African National Research Foundation.

Received: January 2009. Revised: March 2009.

References

[1] A. Csaszar, Generalized topology, generalized continuity, Acta Math. Hungar., 96 (2oo2), 351-

357.

[2] A. Csaszar, δ- and θ- modifications of generalized topologies, Acta. Math. Hungar., to appear.