SUPRA R COMPACTNESS AND SUPRA R … with the new concepts of supra compact, supra Lindelof,...

DOI:10.23883/IJRTER.2018.4058.4RNLX 426

SUPRA – R – COMPACTNESS AND

SUPRA – R – CONNECTEDNESS

Raja Mohammad Latif Department Of Mathematics & Natural Sciences,Prince Mohammad Bin Fahd University

P.O. Box No. 1664 Al Khobar 31952 Saudi Arabia

Abstract—In 2016 El – Shafei, Abo – elhamayel and Al – Shami introduced and investigated a new class of

generalized supra open sets called supra R open set and new separation axioms called iSR T for

i ,1 2 by using R open sets as well as supra R continuous maps, supra R open maps, supra

R closed maps and supra R homeomorphism maps depending on the concepts of supra R open sets. In

this paper, we originally originate the notions of supra R compact spaces and interpret its several effects

and characterizations. Also we newly originate and study the concepts of supra R Lindelof spaces,

countably supra R compact spaces and supra R connected spaces.

Keywords—Supra Topological Space, Supra R Compact Space, Supra R Lindelof Space, Countably

Supra R Compact Space, Supra R Connected Space.

2010 Mathematics Subject Classification: 54B05, 54C20, 54D30.

I. INTRODUCTION

In 1983, A. S. Mashhour et al. [28] introduced the notion of supra topological space and studied

S continuous maps and *S continuous maps. In 2008, R. Devi et al [10] introduced and studied a class of

sets and maps between topological spaces called supra open and supra S continuous maps, respectively. In

2016 M. E. El – Shafei et al [12] introduced the concept of supra R open set, and supra R continuous

functions and investigated several properties for these classes of maps. Recently Krishnaveni and

Vigneshwaran [16] came out with supra bT closed sets and defined their properties. In 2013, Jamal M.

Mustafa [29] came out with the concept of supra b compact and supra b Lindelof spaces. Now we bring

up with the new concepts of supra R compact, supra R Lindelof, countably supra R compact and supra

R connected spaces and present several properties and characteristics of these concepts. Throughout this

paper, ,X and ,Y (or simply, X and Y ) denote topological spaces on which no separation axioms

are assumed unless explicitly stated. For a subset A of a topological space , ,X Cl A , Int A and

X A denote the closure of ,A the interior of A and the complement of A in ,X respectively.

II. PRELIMINARIES

DEFINITION 2.1. Let X be a nonempty set and let * P X A: A X . Then * is called supra

topology on X if *,

*X and

* *. U The pair *X, is called a supra

topological space. Each element *

A is called a supra open set in *X, and C

A X C is called a

supra closed set in *X, .

DEFINITION 2.2. Let *X, be a supra topological space. The supra closure of a set A is denoted by

Supra Cl A and is defined by Supra Cl A

B X : B is a supra closed set in X and A B . I The supra interior of a set A is denoted by

International Journal of Recent Trends in Engineering & Research (IJRTER) Volume 04, Issue 01; January - 2018 [ISSN: 2455-1457]

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Supra Int A and is defined by Supra Int A

U X :U is a supra open set in X and U A . U

DEFINITION 2.3. Let ,X be a topological space and * be a supra topology on X . We call * a supra

topology associated with if * .

DEFINITION 2.4. Let *X, be a supra topological space. A subset A of X is called supra R open

set if there exists a non – empty supra open set G such that G Supra Cl A . The complement of a

supra R open set is called a supra R closed set.

The collection of all supra R open sets in a supra topological space *X, is denoted by

*SRO X, .

THEOREM 2.5. Let *X, be a supra topological space. Then every supra open set in X is supra

R open set in X .

PROOF. Let A be a supra open set in X . Then *

A and A Supra Cl A . Hence it follows that

A is supra R open set.

The converse of the above theorem need not be true as shown by the following example.

EXAMPLE 2.6. Suppose X , , , , 1 2 3 4 5 and have a supra topology *

, , , , , , , , X . 13 2 3 12 3 The set *,3 so the set 3 is not a supra open set in *X, .

Now since *, , 1 2 3 and , , Supra Cl X. 12 3 3 Therefore it follows that 3 is a supra

R open set in *X, .

DEFINITION 2.7. Let *X, be a supra topological space. Then a subset A of X is called a supra semi-

open set if A Supra Cl Supra Int A .

DEFINITION 2.8. Let *X, be a supra topological space. Then a subset A of X is called a supra b

open set if A Supra Cl Supra Int A Supra Int Supra Cl A . U

THEOREM 2.9. Every supra semi-open set in a supra topological space *X, is a supra b open set.

PROOF. Let A be a supra semi-open set in *X, . Then A Supra Cl Supra Int A . Hence,

A Supra Cl Supra Int A Supra Int Supra Cl A U and A is supra b open set in

*X, .


EXAMPLE 2.10. Let *X, be a supra topological space, where X , , 1 2 3 and *

, , , , , , X . 1 1 2 2 3 Hence ,1 3 is a supra b open set, but it is not supra semi-open.

DEFINITION 2.11. Let *X, be a supra topological space. A set A is called supra I open set if

A Supra Int Supra Cl A . The complement of a supra I open set is called a supra

I closed set.

THEOREM 2.12. Let *X, be a supra topological space. Then every supra open set in X is supra

I open set in X .



PROOF. Let A be a supra open set in X . Then

A Supra Cl A , so Supra Int A Supra Int Supra Cl A . Since*A , so

Supra Int A A. Therefore A Supra Int Supra Cl A . Hence it follows that A is supra

I open set.


EXAMPLE 2.13. Suppose X , , , , 1 2 3 4 5 and have the supra topology *

, , , , , , , , X . 13 2 3 12 3 The set *,3 so the set 3 is not a supra open set in *X, .

Now since it clearly follows that Supra Int Supra Cl 3 Supra Int X X. Therefore it

follows that 3 is a supra I open set in *X, .

By the next two examples, we show that neither a supra I open set may be a supra semi-open set nor a

semi-open set may be a supra I open set in a supra topological space.

EXAMPLE 2.14. Suppose X , , , 1 2 3 4 and have the supra topology *

, , , , , , , , X . 13 2 3 12 3 Let A , , . 1 2 4 Then clearly Supra Cl A X. Hence

A Supra Int Supra Cl A Supra Int X X. It shows that A is a supra I open set in X .

Since A Supra Cl Supra Int A Supra Cl . It follows that A is not a supra semi-

open set in X .

EXAMPLE 2.15. Let X a, b, c and * X, , a , b , a,b be a supra topology on X . Then

b, c is a supra semi-open set, but not a supra I open set.

DEFINITION 2.16. Let *X, be a supra topological space. Then a subset A of X is called a supra

open set if A Supra Int Supra Cl Supra Int A .

THEOREM 2.17. Every supra open set in a supra topological space *X, is a supra semi-open set.

PROOF. Let A be a supra open set in *X, . Then we observe that

A Supra Int Supra Cl Supra Int A Supra Cl Supra Int A . Hence, it follows

that A Supra Cl Supra Int A . Thus A is a supra semi-open set in *X, .

The following example shows that the converse need not be true.

EXAMPLE 2.18. Let *X, be a supra topological space, where X , , , 1 2 3 4 and *

, , , , , X . 1 2 12 Hence ,2 3 is a supra semi-open set, but it is not supra open.

THEOREM 2.19. Every supra b open set in a supra topological space *X, is a supra R open set.

PROOF. Let A be a supra b open set in *X, . Then we observe that

A Supra Cl Supra Int A Supra Int Supra Cl A Supra Cl A . U

Case I. Suppose that Supra Int A . Then G Supra Int A is a nonempty supra open set in X

such that G Supra Cl A . This implies that A is a supra R open set in X .

Case II. Suppose that Supra Int A . Then it follows that

A Supra Int A Supra Cl A Supra Cl A . So H upra Int A Supra Cl A



is a nonempty supra open set in X such that H Supra Cl A . This shows that A is a supra R – Open

set in X .

The following example shows that the converse of the above theorem need not be true in general.

EXAMPLE 2.20. Let *X, be a supra topological space, where X , , 1 2 3 and *

, , , , , , , X . 1 2 12 2 3 Then ,1 3 is a supra R open set, but it is not a supra b open set.

The relationships that discussed in the previous theorems and examples are illustrated in the following Figure.

Supra open Supra open Supra semi open Suprab open Supra R open

Now, it is easy to prove the following two propositions.

PROPOSITION 2.21. Every supra neighborhood of any point in a supra topological space *X, is a supra

R open set.

PROPOSITION 2.22. If A is a supra R open set in supra topological space *X, , then every proper

superset of A is supra R open.

The converse of the previous two propositions need not be true as shown in the following example.

EXAMPLE 2.23. Let the supra topology * , , , , , , X 12 13 4 on X , , , . 1 2 3 4 Then

i 1 is supra R open, but is not supra neighborhood of any point.

ii For any proper superset A of ,2 A is supra R open. But 2 is not supra R open.

THEOREM 2.24. Let *X, be a supra topological space. Then the union of an arbitrary family of supra

R open sets is a supra R open set.

PROOF. Let iA :i I be a family of supra R open sets. Then there exist *

i I and *

G such that

*i ii I

G supra Cl A supra Cl A .

U Therefore ii I

AU is a supra R open set.

The intersection of a finite family of supra R open sets may not be supra R open as shown in the

following example.

EXAMPLE 2.25. Let X , , 1 2 3 and * , , , , , , X 13 2 3 be a supra topology on X . Now,

,1 3 and ,1 2 are supra R open sets, but the intersection of them 1 is not a supra R open set.

THEOREM 2.26. Let *X, be a supra topological space. Then the intersection of an arbitrary family of

supra R closed sets is a supra R closed set.

PROOF. Let iA :i I be a family of supra R closed sets. Then C

iiX A :i IA is a family of

supra R open sets. Therefore C

iiX A :i IA U is a supra R open set. Since

C

i iiX X A :i I A :i I .A

U I Hence iA :i II is supra R closed.

The union of a finite family of supra R closed sets may not be supra R closed set as shown in the

following example.

EXAMPLE 2.27. Let X , , 1 2 3 and * , , , , , , X 12 2 3 be a supra topology on X . Now,

2 and 3 are supra R closed sets, but the union of them ,2 3 is not a supra R closed set.

DEFINITION 2.28. Let *X, be a supra topological space and let A be subset of X . Then

i The supra R interior of A denoted by Supra R Int A is the union of all supra R open sets

contained in A.



ii The supra R closure of A denoted by Supra R Cl A is the intersection of all supra R closed

sets containing A.

THEOREM 2.29. Let *X, be a supra topological space. Let A and B be subsets of X . Then the

following statements are true.

(i) Supra-R-Cl AA and Supra-R-Cl AA if and only if A is a supra R closed set.

(ii) Supra-R-Int A A and Supra-R-Int AA if and only if A is a supra R open set.

Supra-R-Int A Supra-R-Cl X Aiii X .

Supra-R-Cl A Supra-R-Int X Aiv X .

(v) Supra-R-Int A Supra-R-Int B Supra-R-Int A B .U U

(vi) Supra-R-CL A B Supra-R-Cl A Supra-R-Cl B .I I

(vii) Supra-R-Int A x X:There exists a supra R open subset U such that x U A .

(viii) Supra-R-Cl A x X:U A , for every supra R open subset U containing x. . I

We will denote X, , Y , and Z , to be topological spaces and their associated supra topological

spaces with *X, , *Y, and *Z, respectively, where * , * and

*.

In the following, we provide definitions of different types of continuous functions as introduced by Elshafei in

2016.

DEFINITION 2.30. A function * *f : X, Y, is called a supra continuous function if the

inverse image of each supra open subset of Y is a supra open subset of X .

DEFINITION 2.31. A function *f : X, Y, is called a supra R continuous function if the

inverse image of each open subset of Y is a supra R open subset of X .

DEFINITION 2.32. A function * *f : X, Y, is called a supra *R continuous function if the

inverse image of each supra open subset of Y is a supra R open subset of X .

DEFINITION 2.33. A function * *f : X, Y, is called a supra R irresolute function if the

inverse image of each supra R open subset of Y is a supra R open subset of X .

DEFINITION 2.34. A map *f : X, Y, is said to be supra R open if the image of every

open subset of X is a supra R open subset of Y.

DEFINITION 2.35. A map *f : X, Y, is said to be supra R closed if the image of every

closed subset of X is a supra R closed subset of Y.

DEFINITION 2.36. Let X , and Y , be two topological spaces and let * and

* be associated

supra topologies with and respectively. A bijective function f : X, Y , is said to be

supra R homeomorphism if f is supra R continuous and supra R open.

DEFINITION 2.37. Let X , and Y , be two topological space and let * and

* be associated supra

topologies with and respectively. Then a function * *f : X, Y, is called strongly supra

R continuous if the inverse image f V1 of every supra R closed set V in Y is supra closed in X .

DEFINITION 2.38. Let X , and Y , be two topological spaces and let * and

* be associated

supra topologies with and respectively. Then a function f : X, Y , is called perfectly



supra R continuous if the inverse image f V1 of every supra R closed set V in Y is both supra

closed and supra open in X .

THEOREM 2.39. Every continuous function is supra R continuous function.

PROOF. Let X , and Y , be two topological spaces and let * and * be associated supra

topologies with and respectively. Let f : X, Y , be a continuous function. Therefore

f A1 is an open set in X for each open set A in Y. But,

* is associated with . That is * . This

implies f A1 is a supra open set in X. Since supra open is supra R open, this implies f A1

is supra

R open in X . Hence f is supra R continuous function.

The converse of the above theorem is not true as shown in the following example.

EXAMPLE 2.40. Let X a, b, c and X, , a,b be a topology on X. The supra topology * is

defined as follows, * X, , a , a,b . Suppose that f : X, X, is a function defined

as follows: f a b, f b c, f c a. The inverse image of the open set a, b is a, c which is

not an open set but it is supra R open. Then f is supra R continuous but it is not continuous.

III. STRONGLY AND PERFECTLY SUPRA R – CONTINUOUS FUNCTIONS In this paper, we introduce a new class of functions called strongly supra continuous and perfectly supra

R continuous functions and investigate their some characterizations.

DEFINITION 3.1. Let X, and Y , be two topological spaces and let * and


topologies respectively such that * and

*. A function * *f : X, Y, is called

strongly supra R continuous if the inverse image of every supra R open subset of Y is supra open in

X .

THEOREM 3.2. A function * *f : X, Y, is strongly supra R continuous if and only if the

inverse image of every supra R closed set in Y is supra closed in X .

PROOF. Assume that f is strongly supra R continuous. Let F be any supra R closed set in Y. Then

CF Y F is supra R open set in Y. Since f is strongly supra R continuous, Cf F1 is supra

open in X . But Cf F X f F 1 1 and so Cf F1 is supra closed in X .

Conversely, assume that the inverse image of every supra R closed set in Y is supra closed in X . Let G

be any supra R open set in Y . Then C

G Y G is supra R closed set in Y . By assumption,

Cf G1 is supra closed in X . But Cf G X f G 1 1 and so f G1

is supra open in X .

Therefore f is strongly supra R continuous.

DEFINITION 3.3. Let X, and Y , be two topological spaces and let * and


topologies respectively such that with * and

*. A function * *f : X, Y, is called

perfectly supra R continuous if the inverse image of every supra R open subset of Y is both supra open

and supra closed in X .

THEOREM 3.4. Let X, and Y , be two topological spaces and let * and


topologies respectively such that * and

*. Suppose that a function * *f : X, Y, is

perfectly supra R continuous. Then f is strongly supra R continuous.



PROOF. Assume that f is perfectly supra R continuous. Let G be any supra R closed set in Y .

Since f is perfectly supra R continuous, f G1 is supra closed in X . Therefore f is strongly supra

R continuous.

The converse of the above theorem need not be true as seen from the following example.

EXAMPLE 3.5. Suppose that X a, b, c , Y , , , 1 2 3 * , a , a, b , X and

* , , , , , , , Y . 1 3 12 13 Let * *f : X, Y, be a function defined by

f a ,1 f b 2 and f c .3 Then f is strongly supra R continuous but not perfectly supra

R continuous, since for the supra R closed set V , 2 3 in Y, f V f , b, c 1 1 2 3 is

not both supra open and supra closed in X .

THEOREM 3.6. A function * *f : X, Y, is perfectly supra R continuous if and only if the

inverse image of every supra R closed set in Y is both supra open and supra closed in X .

PROOF. Assume that f is perfectly supra R continuous. Let F be any supra R closed set in Y . Then

CF Y F is supra R open set in Y . Since f is perfectly supra R continuous, Cf F1 is both

supra open and supra closed in X . But Cf F X f F 1 1 and so f F1 is both supra open and

supra closed in X .

Conversely, assume that the inverse image of every supra R closed set in Y is both supra open and supra

closed in X . Let G be any supra R open set in Y. Then C

G Y G is supra R closed set in Y . By

assumption, Cf G1 is supra closed in X . But Cf G X f G 1 1 and so f G1 is both supra

open and supra closed in X . Therefore f is perfectly supra R continuous.

THEOREM 3.7. If a function * *f : X, Y, is strongly supra R continuous then it is

R irresolute but not conversely.

PROOF. Let * *f : X, Y, be strongly supra R continuous function. Let F be a supra

R closed set in Y. Since f is strongly supra R continuous, f F1 is supra closed in X. Since every

supra open set is supra R open, so every supra closed set is supra R closed set, f F1 is

R closed in X. Hence f is irresolute.


EXAMPLE 3.8. Let X a, b, c , Y , , , 1 2 3 * , a , X and * , , , , Y . 1 12 Let

* *f : X, Y, be a function defined by f a ,1 f b 2 and f c .3 Then f is

R irresolute but not strongly supra R continuous, since for the supra R closed set V 2 in Y,

f V f b 1 1 2 is not supra closed in X .

THEOREM 3.9. If a function * *f : X, Y, is perfectly supra R continuous then it is

R irresolute but not conversely.

PROOF. Let * *f : X, Y, be perfectly supra R continuous function. Let G be a supra

R open set in Y. Since f is perfectly supra R continuous, f G1 is both supra open and supra closed

in X . Since every supra open set is supra R open set and so f G1 is supra R open in X. Hence f

is R irresolute.




EXAMPLE 3.10. Let X a, b, c , Y , , , 1 2 3 * , a , X and * , , , , Y . 1 12 Let

* *f : X, Y, be a function defined by f a ,1 f b 2 and f c .3 Then f is

R irresolute but not perfectly supra R continuous, since for the supra R closed set V 2 in Y,

f V f b 1 1 2 is not both supra open and supra closed in X .

IV. SUPRA R COMPACTNESS

DEFINITION 4.1. A collection iA :i I of supra R open sets in a supra topological space *X, is

called a supra R open cover of a subset B of X if iB A :i I U holds.

DEFINITION 4.2. A supra topological space *X, is called supra R compact if every supra

R open cover of X has a finite subcover.

DEFINITION 4.3. A subset B of a supra topological space *X, is said to be supra R compact

relative to *X, if, for every collection iA :i I of supra R open subsets of X such that

iB A :i I U there exists a finite subset I 0 of I such that i

B A :i I . 0U

DEFINITION 4.4. A subset B of a supra topological space *X, is said to be supra R compact if B

is supra R compact as a subspace of X .

THEOREM 4.5. Every supra R compact space is supra compact.

PROOF. Let iA :i I be a supra open cover of X, . Since * *SRO X, . So i

A :i I is a

supra R open cover of *X, . Since *X, is supra R compact. So supra R open cover

iA :i I of *X, has a finite subcover say i

A :i , , . . ., n1 2 for X . Hence *X, is a supra

R compact space.

THEOREM 4.6. Every supra R closed subset of a supra R compact space is supra R compact

relative to X .

PROOF. Let A be a supra R closed subset of a supra topological space *X, . Then CA X A is

supra R open in *X, . Let iA :i I be a supra R open cover of A by supra R open

subsets in *X, . Let * C

iA :i I A U be a supra R open cover of *X, . That is

* C

iX A :i I A . U U U By hypothesis *X, is supra R compact and hence

* is reducible

to a finite subcover of *X, say C

nX A A ... A A ; 1 2U U U U k

A for k , , . . ., n. 1 2 But A and

CA are disjoint. Hence nA A A ... A ; 1 2U U U k

A for k , , . . ., n. 1 2 Thus a supra

R open cover of A contains a finite subcover. Hence A is supra R compact relative to *X, .

THEOREM 4.7 A supra R continuous image of a supra R compact space is compact.

PROOF. Let X , and Y , be two topological spaces and let * be associated supra topology with .

Let *f : X, Y, be a supra R continuous map from a supra R compact space X onto a

topological space Y. Let iA :i I be an open cover of Y . Then 1 :

if A i I is a supra R open

cover of X, as f is supra R continuous. Since X is supra R compact, the supra R open cover of X,

1 :i

f A i I has a finite subcover say 1 : 1, 2, . . ., .i

f A i n Therefore



1 : 1, 2, . . ., ,i

X f A i n which implies : 1,2,..., ,i

f X A i n then

: 1,2,..., .i

Y A i n That is : 1,2,...,i

A i n is a finite subcover of iA :i I for .Y Hence Y

is compact.

THEOREM 4.8 A supra *R continuous image of a supra R compact space is supra compact.

PROOF. Let X , and Y , be two topological spaces and let * and


topologies with and respectively. Let * *f : X, Y, be a supra *R continuous map

from a supra R compact space X onto a supra topological space Y. Let iA :i I be a supra open

cover of Y . Then 1 :i

f A i I is a supra R open cover of X , as f is supra *R continuous.

Since X is supra R compact, the supra R open cover of X, 1 :i

f A i I has a finite subcover

say 1 : 1, 2, . . .,i

f A i n . Therefore 1 : 1, 2, . . ., ,i

X f A i n which implies that

: 1, 2, . . . , ,i

f X A i n then : 1, 2, . . . , .i

Y A i n That is : 1, 2, . . . ,i

A i n is a

finite subcover of iA :i I for .Y Hence Y is supra compact.

THEOREM 4.9. Let X , and Y , be two topological spaces and let * and


topologies with and respectively. If a map * *f : X, Y, is R irresolute and a subset

S of X is supra R compact relative to X, , then the image f S is supra R compact relative

to Y , .

PROOF. Let iA :i I be a collection of supra R open subsets of Y , , such that

if S A : i I . U Since f is R irresolute. So i

S f A :i I , 1U where

if A :i I SRO X, . 1 Since S is supra R compact relative to X , , there exists a finite

subcollection 1 2, , . . .,

nA A A such that i

S f A :i , , . . ., n . 1 12U That is

nf S A ,A ,...,A . 1 2U Hence f S is supra R compact relative to Y , .

THEOREM 4.10. Suppose that a map * *f : X, Y, is strongly supra R continuous map

from a supra compact space *X, onto a supra topological space *Y, , then *Y, is supra

R compact.

PROOF. Let iA :i I be a supra R cover of Y , . Since f is strongly supra R continuous,

if A :i I 1 is a supra open cover of X, . Again, since X, is supra compact, the supra open

cover if A :i I 1 of X, has a finite sub cover say i

f A :i , , . . . , n . 1 12 Therefore

iX f A :i , , . . ., n , 1 12U which implies i

f X A : i , , . . .,n , 1 2U so that

iY A : i , , . . ., n . 1 2U That is n

A , A , . . . , A1 2 is a finite subcover of iA :i I for Y , .

Hence Y , is supra R compact.

THEOREM 4.10. Suppose that a map f : X, Y , is perfectly supra R continuous map from

a supra compact space X , onto a supra topological space Y , . Then Y , is supra

R compact.



PROOF. Let iA :i I be a supra R open cover of Y , . Since f is perfectly supra

R continuous, if A :i I 1 is a supra open cover of X , . Again, since X , is supra

compact, the supra open cover if A :i I 1 of X , has a finite sub cover say

if A :i , , . . ., n . 1 12 Therefore i

X f A :i , , . . .,n , 1 1 2U which implies

if X A : i , ,. . .,n , 1 2U so that i

Y A : i , , . . ., n . 1 2U That is nA , A , . . ., A1 2 is a

finite sub cover of iA :i I for Y , . Hence Y , is supra compact.

THEOREM 4.11. Let f : X, Y , be supra R irresolute map from supra R compact

space X , onto supra topological space Y , , then Y , is supra R compact.

PROOF. Let f : X, Y , be supra R irresolute map from a supra R compact space

X, onto a supra topological space Y , . Let iA :i I be a supra R open cover of Y , .

Then if A :i I 1 is a supra R open cover of X, , since f is supra R irresolute. As

X , is supra R compact, the supra R open cover if A :i I 1 of X, has a finite sub

cover say if A :i , , . . ., n . 1 12 Therefore i

X f A :i , , . . ., n , 1 12U which implies

if X A : i , ,...,n , 12U so that i

Y A : i , , . . ., n . 1 2U That is nA , A ,. . ., A1 2 is a finite

sub cover of iA :i I for Y , . Hence Y , is supra R compact.

THEOREM 4.12. If X , is supra compact and every supra R closed set in X is also supra closed in

X, then X , is supra R compact.

PROOF. Let iA :i I be a supra R open cover of X. Since every supra R closed set in X is also

supra closed in X . Thus iX A :i I is a supra closed cover of X and hence i

A :i I is a supra

open cover of X, Since X , is supra compact. So there exists a finite sub cover iA : i , , . . ., n1 2

of iA :i I such that i

X A : i , , . . ., n . 1 2U Hence X , is a supra R compact space.

THEOREM 4.13. A supra topological space X , is supra R compact if and only if every family of

supra R closed sets of X , having finite intersection property has a non empty intersection.

PROOF. Suppose X, is supra R compact. Let iA :i I be a family of supra R closed sets

with finite intersection property. Suppose i

i I

A ,

I then iX A :i I X. I This implies

iX A :i I X. U Thus the cover i

X A :i I is a supra R open cover of X , . Then,

the supra R open cover iX A :i I has a finite sub cover say i

X A :i , , . . .,n . 12 This

implies that iX X A :i , , . . .,n 12U which implies i

X X A :i , , . . .,n , 1 2I which

implies iX X A :i , , . . .,n , 1 2I which implies i

A :i , , . . .,n . 1 2I This disproves the

assumption. Hence iA :i , , . . .,n . 1 2I

Conversely suppose X, is not supra R compact. Then there exits a supra R open cover of

X, say iG :i I having no finite sub cover. This implies for any finite sub family

iG :i , , . . .,n1 2 of i

G :i I , we have iG :i , , . . .,n X, 1 2U which implies



iX G :i , , . . .,n X X, 12U hence i

G :i , , . . .,n . 1 2I Then the family

iX G :i I of supra R closed sets has a finite intersection property. Also by assumption

iX G :i , ,...,n 12I which implies i

X G :i , ,...,n , 12U so that

iG :i , , . . .,n X. 1 2U This implies i

G :i I is not a cover of X, . This disproves the fact that

iG :i I is a cover for X , . Therefore a supra R open cover i

G :i I of X , has a finite

sub cover iG :i , , . . ., n .1 2 Hence X , is supra R compact.

THEOREM 4.14. Let A be a supra R compact set relative to a supra topological space X and B be a

supra R closed subset of X . Then A BI is supra R compact relative to X .

PROOF. Let A be supra R compact relative to X . Let iA :i I be a cover of A BI by supra

R open sets in X . Then C

iA :i I B U is a cover of A by supra R open sets in X, but A is

supra R compact relative to X, so there exist ni , i , . . ., i I1 2 such that

j

C

iA A : j , , ...,n B . 12U U Then it follows that ji

A B A B : j , , . . ., n 1 2I U U I

ji

A : j , , . . ., n .1 2U Hence A BI is supra R compact relative to X .

THEOREM 4.15. Suppose that a function f : X, Y , is supra R irresolute and a subset of X

is supra R compact relative to X . Then f B is supra R compact relative to Y .

PROOF. Let iA :i I be a cover of f B by supra R open subsets of Y . Since f is supra

R irresolute. Then if A :i I 1 is a cover of B by supra R open subsets of X . Since B is

supra R compact relative to X, if A :i I 1 has a finite sub cover say

nf A , f A , . .., f A 1 1 1

1 2 for B. Now i

A :i , ,. . ., n1 2 is a finite sub cover of

iA :i I for f B . So f B is supra R compact relative to Y.

V. COUNTABLY SUPRA R COMPACTNESS

In this section, we concentrate on the concept of countably supra R compactness and their properties.

DEFINITION 4.1. A supra topological space X, is said to be countably supra R compact if every

countable supra R open cover of X has a finite sub cover.

THEOREM 4.2. If X, is a countably supra R compact space, then X, is countably supra

compact.

PROOF. Let X, be countably supra R compact space. Let iA :i I be a countable supra open

cover of X, . Since * SRO X, . So iA :i I is a countable supra R open cover of

X, . Since X, is countably supra R compact, so the countable supra R open cover

iA :i I of X, has a finite sub cover say i

A :i , , . . .,n1 2 for X . Hence X, is a countably

supra compact space.

THEOREM 4.3. If X, is countably supra compact and every supra R closed subset of X is supra

closed in X, then X, is countably supra R compact.



PROOF. Let X, be countably supra compact space. Let iA :i I be a countable supra R open

cover of X, . Since every supra R closed subset of X is supra closed in X. Thus every supra

R open set in X is supra open in X . Therefore iA :i I is a countable supra open cover of X, .

Since X, is countably supra compact, countable supra open cover iA :i I of X, has a finite

sub cover say iA :i , , . . .,n1 2 for X . Hence X, is a countably supra R compact space.

THEOREM 4.4. Every supra R compact space is countably supra R compact.

PROOF. Let X, be supra R compact space. Let iA :i I be a countable supra R open cover

of X, . Since X, is supra R compact, so supra pen cover iA :i I of X, has a finite

subcover say iA :i , , . . .,n1 2 for X, . Hence X, is countably supra R compact space.

THEOREM 4.5. Let f : X, Y , be a supra R continuous injective mapping. If X is

countably supra R compact space, then Y , is countably compact.

PROOF. Let f : X, Y , be a supra R continuous map from a countably supra

R compact space X, onto a topological space Y , . Let iA :i I be a countable open cover

of Y. Then if A :i I 1 is a countable supra R open cover of X, as f is supra R continuous.

Since X is countably supra R compact, the countable supra R open cover if A :i I 1 of X

has a finite sub cover say if A :i , ,...,n . 1 12 Therefore i

X f A :i , , . . ., n , 1 12U which

implies that if X A : i , , . . .,n , 1 2U then i

Y A : i , ,. . ., n . 1 2U That is

iA : i , , . . ., n1 2 is a finite sub cover of i

A :i I for Y . Hence Y is countably compact.

THEOREM 4.6. Suppose that a map f : X, Y , is perfectly supra R continuous map from a

countably supra compact space X , onto a supra topological space Y , . Then Y , is countably

supra R compact.

PROOF. Let iA :i I be a countable supra R open cover of Y , . Since f is perfectly supra

R continuous, if A :i I 1 is a countable supra open cover of X , . Again, since X , is

countably supra compact, the countable supra open cover if A :i I 1 of X , has a finite sub cover

say if A :i , , . . .,n . 1 1 2 Therefore i

X f A :i , , . . .,n , 1 1 2U which implies


Y A : i , , . . ., n . 1 2U That is nA ,A , . . ., A1 2 is a finite

sub cover of iA :i I for Y , . Hence Y , is countably supra R compact.

THEOREM 4.7. Suppose that a map f : X, Y , is strongly supra R continuous map from a

countably supra compact space X , onto a supra topological space Y , . Then Y , is countably

supra R compact.

PROOF. Let iA :i I be a countable supra R open cover of Y , . Since f is strongly supra

R continuous, if A :i I 1 is a countable supra open cover of X , . Again, since X , is

countably supra compact, the countable supra open cover if A :i I 1 of X , has a finite sub cover

say if A :i , ,...,n . 1 12 Therefore i

X f A :i , ,...,n , 1 12U which implies




Y A : i , ,...,n . 12U That is nA ,A ,...,A1 2 is a finite sub

cover of iA :i I for Y , . Hence Y , is countably supra R compact.

THEOREM 4.8. The image of a countably supra R compact space under a supra R irresolute map is

countably supra R compact.

PROOF. Suppose that a map f : X, Y , is a supra R irresolute map from a countably supra

R compact space X , onto a supra topological space ( Y , . Let iA :i I be a countable supra

R open cover of Y , . Then if A :i I 1 is a countable supra R open cover of X, ,

since f is supra R irresolute. As X, is countably supra R compact, the countable supra

R open cover if A :i I 1 of X , has a finite sub cover say i

f A :i , , . . ., n . 1 12

Therefore iX f A :i , , . . ., n , 1 12U which implies i

f X A : i , ,...,n , 12U so that

iY A : i , , . . ., n . 1 2U That is n

A , A , . . ., A1 2 is a finite sub cover of iA :i I for Y , .

Hence Y , is countably supra R compact.

V. SUPRA R LINDELOF SPACE

In this section, we concentrate on the concept of supra R Lidelof space and their properties.

DEFINITION 5.1. A supra topological space X, is said to be supra R Lindelof space if every supra

R open cover of X has a countable sub cover.

THEOREM 5.2. Every supra R Lindelof space is supra Lindelof space.

PROOF. Let *X, be a supra R Lindelof space. Let iA :i I be a supra open cover of *X, .

Since * *SRO X, . Therefore iA :i I is a supra R open cover of *X, . Since *X, is

supra R Lindelof space, the supra R open cover iA :i I of *X, has a countable sub cover

say iA :i I 0 for X, for some countable subset I 0 of I. Hence *X, is a supra Lindelof space.

THEOREM 5.3. If *X, is supra R Lindelof space, then X, is Lindelof space.

PROOF. Let iA :i I be an open cover of X . Since * *SRO X, . Therefore i

A :i I is

a supra R open cover of *X, . Since *X, is supra R Lindelof, supra R open cover

iA :i I of *X, has a countable sub cover say i

A :i I 0 for X, for some countable subset I 0 of

I. Hence *X, is a Lindelof space.

THEOREM 5.4. Every supra R compact space is supra R Lindelof space.

PROOF. Let *X, be a supra R compact soace. Let iA :i I be a supra R open cover of

*X, . Since X, is supra R compact space. Then iA :i I has a finite sub cover say

iA :i , ,. . ., n .1 2 Since every finite sub cover is always countable sub cover and therefore

iA :i , ,. . ., n1 2 is countable sub cover of i

A : i I . Hence X, is supra R Lindelof space.

THEOREM 5.5. A supra R continuous image of a supra R Lindelof space is supra Lindelof space.

PROOF. Let * *f : X, Y, be a supra R continuous map from a supra R Lindelof space

X onto a supra topological space Y. Let iA :i I be a supra open cover of Y. Then i

f A :i I 1



is a supra R open cover of X, as f is supra R continuous. Since X is supra R Lindelof space,

the supra R open cover if A :i I 1 of X has a countable sub cover say i

f A :i I 1

0 for

some I I0 and I0 is countable. Therefore iX f A :i I , 1

0U which implies

if X A : i I , 0U then i

Y A : i I . 0U That is iA :i I 0 is a countable sub cover of

iA :i I for Y. Hence Y is is a supra Lindelof space.

THEOREM 5.6. The image of a supra R Lindelof space under a supra R irresolute map is supra

R Lindelof space.

PROOF. Suppose that a map * *f : X, Y, be supra R irresolute map from a supra

R Lindelof space *X, onto a supra topological space *Y, . Let iA :i I be a supra

R open cover of *Y, . Then if A :i I 1 is a supra R open cover of *X, . Since f is

supra R irresolute. As *X, is supra R Lindelof space, the supra R open cover

if A :i I 1 of *X, has a countable sub cover say i

f A :i I 1

0 for some I I0 and I0

is countable. Therefore iX f A :i I , 1

0U which implies if X A : i I , 0U so that

iY A : i I . 0U That is i

A :i I 0 is a countable sub cover of iA :i I for Y. Hence *Y, is

a supra R Lindelof space.

THEOREM 5.7. If *X, is supra R Lindelof space and countably supra R compact space, then

*X, is supra R compact space.

PROOF. Suppose *X, is supra R Lindelof space and countably supra R compact space. Let

iA :i I be a supra R open cover of *X, . Since *X, is supra R Lindelof space,

iA :i I has a countable sub cover say i

A :i I 0 for some I I0 and I0 is countable. Therefore

iA :i I 0 is a countable supra R open cover of *X, . Again, since *X, is countably supra

R compact space, iA :i I 0 has a finite sub cover and say i

A :i , , . . ., n .1 2 Therefore

iA :i , , . . .,n1 2 is a finite sub cover of i

A :i I for *X, . Hence *X, is a supra

R compact space.

THEOREM 5.8. If a function * *f : X, Y, is supra R irresolute and a subset of X is supra

R Lindelof relative to X, then f B is supra R Lindelof relative to Y.

PROOF. Let iA :i I be a cover of f B by supra R open subsets of Y. By hypothesis f is supra

R irresolute and so if A :i I 1 is a cover of B by supra R open subsets of X . Since B is

supra R Lindelof relative to X, if A :i I 1 has a countable sub cover say i

f A :i I 1

0 for

B, where I0 is a countable subset of I . Now iA :i I 0 is a countable sub cover of i

A :i I for

f B . So f B is supra R Lindelof relative to Y.



VI. SUPRA R CONNECTEDNESS

DEFINITION 6.1. A supra topological space *X, is said to be supra connected if X cannot be written as

a disjoint union of two nonempty supra open sets. A subset of *X, is supra connected if it is supra

connected as a subspace.

DEFINITION 6.2. A supra topological space *X, is said to be supra R connected if X cannot be

written as a disjoint union of two nonempty supra R open sets. A subset of X , is supra

R connected if it is supra R connected as a subspace.

THEOREM 6.3. Every supra R connected space is supra connected.

PROOF. Let A and B be two nonempty disjoint proper supra open sets in X . Since every supra open set is

supra R open set. Therefore A and B are nonempty disjoint proper supra R open sets in X and X is

supra R connected space. Therefore X A B. U Therefore X is supra R connected.

The converse of the above theorem need not be true in general, which follows from the following example.

EXAMPLE 6.4. Let X , , , 1 2 3 4 and * , , , , , , X . 1 2 1 2 3 Then *X, is a supra

topological space. Since X cannot be written as a disjoint union of any two nonempty supra open sets.

Therefore *X, is a supra connected topological space. We notice that both 1 and , ,2 3 4 are supra

R open sets in *X, because Supra Int Supra Cl 1 1 Supra Int X X and

, , Supra Int Supra Cl , , 2 3 4 2 3 4 Supra Int X X. Therefore 1 and , ,2 3 4

are nonempty disjoint supra R open sets such that X , , . 1 2 3 4U Hence *X, is not a supra

R connected space.

THEOREM 6.5. Let *X, a supra topological space. Then the following statements are equivalent

(i) *X, is supra R connected.

(ii) The only subsets of *X, which are both supra R open and supra R closed are the empty set

X and .

(iii) Each supra R continuous map of *X, into a discrete space Y , with at least two points is a

constant map.

PROOF. (i)⇒(ii) Let G be a supra R open and supra R closed subset of *X, . Then X G is also

both supra R open and supra R closed. Then X G X G U a disjoint union of two nonempty

supra R open sets which contradicts the fact that *X, is supra R connected. Hence G or

G X.

(ii)⇒(i) Suppose that X A B U where A and B are disjoint nonempty supra R open subsets of

*X, . Since A X B, then A is both supra R open and supra R closed. By assumption A

or A X, which is a contradiction. Hence *X, is supra R connected.

(ii)⇒(iii) Let * *f : X, Y, be a supra R continuous map, where *Y, is discrete space

with at least two points. Then f y1 is supra R closed and supra R open for each y Y. That is

*X, is covered by supra R closed and supra R open covering f y : y Y . 1 By

assumption, f y 1 or f y X 1

for each y∈ Y. If f y 1 for each y Y, then f fails to



be a map. Therefore their exists at least one point say *f y , 1 *y Y such that *f y X. 1 This

shows that f is a constant map.

(iii)⇒(ii) Let G be both supra R open and supra R closed in *X, . Suppose G . Let

* *f : X, Y, be a supra R continuous map defined by f G a and f X G b

where a b and a,b Y. By assumption, f is constant so G X.

THEOREM 6.6. Let *f : X, Y, be a supra R continuous surjection and *X, be supra

R connected. Then Y , is connected.

PROOF. Suppose Y , is not connected. Let Y A B, U where A and B are disjoint nonempty open

subsets in Y , . Since f is supra R continuous, X f A f B , 1 1U where f A1 and

f B1 are disjoint nonempty supra R open subsets in *X, . This disproves the fact that *X, is

supra R connected. Hence Y , is connected.

THEOREM 6.7. If * *f : X, Y, is a supra R irresolute surjection and X is supra

R connected, then Y is supra R connected.

PROOF. Suppose that Y is not supra R connected. Let Y A B, U where A and B are nonempty

supra R open sets in Y. Since f is supra R irresolute and onto, X f A f B , 1 1U where

f A1 and f B1

are disjoint nonempty supra R open sets in *X, . This contradicts the fact that

*X, is supra R connected. Hence *Y, is supra R connected.

THEOREM 6.8. Suppose that every supra R closed set in X is supra closed in X and X is supra

connected. Then X is supra R connected.

PROOF. Suppose that X is supra connected. Then X cannot be expressed as the disjoint union of two

nonempty proper supra open subset of X . Suppose X is not supra R connected space. Let A and B be

any two supra R open subsets of X such that Y A B, U where A BI and A X, B X.

Since every supra R closed set in X is supra closed in X . So every supra R open set in X is supra

open in X . Hence A and B are supra open subsets of X, which contradicts that X is supra connected.

Therefore X is supra R connected.

THEOREM 6.9. If two supra R open sets C and D form a separation of X and if Y is supra

R connected subspace of X, then Y lies entirely within C or D .

PROOF. By hypothesis C and D are both supra R open sets in X . The sets C YI and D YI are supra

R open in Y, these two sets are disjoint and their union is Y . If they were both nonempty, they would

constitute a separation of Y . Therefore, one of them is empty. Hence Y must lie entirely in C or D .

THEOREM 6.10. Let A be a supra R connected subspace of X . If A B Supra R Cl A ,

then B is also supra R connected.

PROOF. Let A be supra R connected. Let A B Supra R Cl A . Suppose that B C D U is

a separation of B by supra R open sets. Thus by previous theorem above A must lie entirely in C or

D . Suppose that A C, then it implies that Supra R Cl A Supra R Cl C . Since

Supra R Cl C and D are disjoint, B cannot intersect D . This disproves the fact that D is

nonempty subset of B. So D which implies B is supra R connected.



ACKNOWLEDGEMENT

The author is highly and gratefully indebted to the Prince Mohammad Bin Fahd University, Saudi Arabia, for

providing research facilities during the preparation of this research paper.

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Dr. Raja Mohammad Latif, Department of Mathematics & Natural Sciences, Prince Mohammad Bin Fahd

University, Al Khbar, Saudi Arabia. [email protected] & [email protected] &

[email protected].

mailto:[email protected]

mailto:[email protected]

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