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 i SR T for i , 12 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 * PX 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 * Ais 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
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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
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
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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, .
The converse of the above theorem need not be true as shown by the following example.
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 .
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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.
The converse of the above theorem need not be true as shown by the following example.
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
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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.
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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
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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
* be associated supra
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
* be associated supra
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
* be associated supra
topologies respectively such that * and
*. Suppose that a function * *f : X, Y, is
perfectly supra R continuous. Then f is strongly supra R continuous.
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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.
The converse of the above theorem need not be true as seen from the following example.
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.
The converse of the above theorem need not be true as seen from the following example.
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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
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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
* be associated supra
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
* be associated supra
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.
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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
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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.
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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
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 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
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if X A : i , ,...,n , 12U so that i
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
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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.
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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
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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.
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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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