D O I: 1 0 .1 5 0 1 / C o m m u a 1 _ 0 0 0 0 0 0 0 7 7 3 IS S N 1 3 0 3 –5 9 9 1
- -CONVERGENCE, - -ACCUMULATION AND
- -COMPACTNESS
ALIAS B. KHALAF AND HARIWAN Z. IBRAHIM
Abstract. The aim of the present paper is to introduce the concept of -compactness by means of -operation de…ned on the family of -open sets of a topological space. We de…ne the - -convergence, - -accumulation points of a …lterbase and give some of their properties. Some characterizations and properties of - -compact spaces are obtained.
1. Introduction
The notion of compactness is useful and fundamental notion of not only general topology but also of other advanced branches of mathematics. Many researchers have investigated the basic properties of compactness. The productivity and fruit-fulness of this notion of compactness motivated mathematicians to generalize this notion. Njastad [5] introduced a new class of generalized open sets in a topologi-cal space topologi-called -open sets. The concept of operation was initiated by Ibrahim [6]. He also introduced the concept of -open sets. The aim of this paper is to introduce the concept of - -compactness in topological spaces and is to give some characterizations of - -compact spaces. The notion of - -convergence and
- -accumulation are de…ned and are used to characterize - -compactness. 2. Preliminaries
Throughout the present paper (X; ) (or simply X) denotes a topological space. Let A be a subset of X. We denote the interior and the closure of a set A by Int(A) and Cl(A), respectively. A subset A of X is said to be -open [5] if A Int(Cl(Int(A))). The complement of an -open set is said to be -closed. The family of all -open sets in a topological space (X; ) is denoted by O(X; ). An operation : O(X; ) ! P (X) [6] is a mapping satisfying the condition, V V for each V 2 O(X; ). We call the mapping an operation on O(X; ). A subset
Received by the editors: Received: Feb. 28, 2016 Accepted: Aug. 10, 2016.
2010 Mathematics Subject Classi…cation. Primary 05C38, 15A15; Secondary 05A15, 15A18. Key words and phrases. Operation, -open set, -open set, - -convergence, -accumulation, - -compact.
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A of X is called an -open set [6] if for each point x 2 A, there exists an -open set U of X containing x such that U A. The complement of an -open set is said to be -closed. We denote the set of all -open (resp., -closed) sets of (X; ) by O(X; ) (resp., C(X; ) ). A point x 2 X is in Cl -closure [6] of a set A X, if U \ A 6= for each -open set U containing x. The Cl -closure of A is denoted by Cl (A). An operation on O(X; ) is said to be -regular [6] if for every -open sets U and V containing a point x 2 X, there exists an -open set W containing x such that W U \ V . An operation on O(X; ) is said to be -open [6] if for every -open set U containing x 2 X, there, exists an -open set V of X such that x 2 V and V U . An operation : O(X) ! P (X) is said to be -monotone [3] if for all A; B 2 O(X), A B implies A B . The operation id : O(X; ) ! P (X) is de…ned by id(V ) = V for any set V 2 O(X; ) this operation is called the identity operation on O(X; ) [6]. An operation : O(X) ! P (X) is said to be -additive [3] if (A [ B) = A [ B for all A; B 2 O(X).
De…nition 1. [1] A topological space (X; ) is said to be -regular if for each x 2 X and for each -open set V in X containing x, there exists an -open set U in X containing x such that U V .
De…nition 2. [2] Let H be any subset of X, jH = fV = U \ H : U 2 O(X)g. An operation from O(X) to P (X) is said to be -stable with respect to H if induces an operation H: jH ! P (H) satisfying the following two properties:
(1) (U \ H) H = U \ H for every U 2 O(X) and
(2) W \ H = S \ H implies that W \ H = S \ H for every W; S 2 O(X). De…nition 3. [4] A space X with an operation on O(X) is called - -T2space if
for any two distinct points x; y 2 X, there exist two -open sets U and V containing x and y, respectively, such that U \ V = .
De…nition 4. [8] A …lterbase F is said to be -converges to a point x 2 X if for each -open set V containing x, there exists an F 2 F such that F V .
De…nition 5. [7] A space X is said to be -compact i¤ every -open cover of X has a …nite subcover
3. - -convergence, - -accumulation and - -compact
De…nition 6. A …lterbase F in a topological space (X; ) with an operation on O(X) is said to be:
(1) - -converges to a point x 2 X if for every -open set V containing x, there exists F 2 F such that F V .
(2) - -accumulates to a point x 2 X if F \ V 6= , for every -open set V containing x and every F 2 F.
Remark 1. If a …lterbase F in a topological space (X; ) is -converges to a point x 2 X, then F is - -converges to a point x 2 X
The converse of above remark is not true in general as it is shown in the following example.
Example 1. Let X = fa; b; cg, = f ; X; fagg be a topology on X and F = ffbg; fb; cgg be a …lterbase. For each A 2 O(X) we de…ne on O(X) by A = Cl(A). Then, F is - -converges to a point a 2 X, but F is not -converges to a. Corollary 1. Let (X; ) be a topological space and be an operation on O(X). If a …lterbase F in X, - -converges to a point x 2 X, then F - -accumulates to x.
The converse of above corollary is not true in general as it is shown in the following example.
Example 2. Let X = fa; b; cg, = f ; X; fag; fbg; fa; bgg be a topology on X and F = ffa; bgg be a …lterbase. For each A 2 O(X) we de…ne on O(X) by
A = Cl(A) if A = fbg
X if A 6= fbg:
Then, F is - -accumulates to a point b 2 X, but F is not - -converges to b. Proposition 1. Let F be a …lterbase in a topological space (X; ), is -open and E is any -open set containing x. If there exists an F 2 F such that F E, then F is - -converges to a point x 2 X.
Proof. Obvious.
Proposition 2. Let F be a …lterbase in a topological space (X; ), is -open and E is any -open set containing x such that F \ E 6= for each F 2 F, then F is
- -accumulation to a point x 2 X.
Proof. The proof is similar to Proposition 1.
Theorem 1. If a …lterbase F in X is contained in a …lterbase which is -accumulate to x 2 X, then F is - --accumulate to x.
Proof. Obvious.
Theorem 2. Let (X; ) be a topological space and an -regular operation on O(X). If a …lterbase F in X is - -accumulates to x, then there exists a …lterbase H in X such that F H and H is - -converges to x.
Proof. Let the …lterbase F be - -accumulate to x. Hence for every -open set U containing x and for each A 2 F, A \ U 6= . Then x 2 Cl (A), for every A 2 F. Consider the set G = fA \ U : U is -open set of X containing x and A 2 Fg. Suppose that G1; G2 2 G. Then G1\ G2 = (A1\ U1) \ (A2\ U2) =
(A1\A2)\(U1\U2) for every A1; A22 F and U1; U2are -open sets of X containing
x. Since is -regular, then there exists an -open set U3of X containing x such
A3 A1\ A2. Hence A3\ U3 G1\ G2. Thus we see that G is a …lterbase. Now
the set H = fB : 9C 2 G with C Bg is …lter generated by G. For each -open set U containing x and for each A 2 F, U A \ U 2 H, where A \ U 2 G. So H is - -converges to x. Also for each A 2 F, A = X \ A 2 G, and thus A 2 H. Hence, F H.
Corollary 2. Let (X; ) be a topological space and be an -monotone operation on O(X). If a maximal …lterbase in X, - -accumulates to a point x 2 X then it
- -converges to x.
Proof. Similar to the proof of Theorem 2.
De…nition 7. A topological space (X; ) is said to be - -compact if for every -open cover fVi : i 2 Ig of X, there exists a …nite subset I0 of I such that
X =SfVi : i 2 I0g.
Theorem 3. Every -compact space is - -compact. Proof. Obvious.
Remark 2. The converse of above theorem is not true in general as it is shown in the following example.
Example 3. Let N be the set of all natural numbers with the discrete topology . For a topological space (N; ), we have, O(N; ) = . Let : O(N; ) ! P (N) be an operation de…ned by A = N for every set A 2 O(N; ). Then, N is
- -compact but not -compact.
Remark 3. If is -identity operation, then - -compactness coincides with -compact.
Theorem 4. If a topological space (X; ) is - -compact for some operation on O(X) such that (X; ) is -regular, then (X; ) is -compact.
Proof. Let U = fUi: i 2 Ig be an -open cover of X. Since X is -regular, then
for each i 2 I, Vi Ui. Since Vi is -open set, therefore the set fVi: i 2 Ig is an
-open cover of X. Since X is - -compact, there is a …nite subset I0 of I such
that X = SfVi : i 2 I0g. For each i 2 I0, there exists Ui such that Vi Ui,
therefore we have X =SfUi: i 2 I0g and so X is -compact.
Theorem 5. Let (X; ) be a topological space, and an -monotone operation on O(X). Then, the following conditions are equivalent:
(1) (X; ) is - -compact.
(2) Each maximal …lterbase in X - -converges to some point of X. (3) Each …lterbase in X - -accumulates to some point of X.
Proof. (1) ) (2): Suppose that X is - -compact space and let F = fFi : i 2 Ig
be a maximal …lterbase. Suppose that F does not - -converges to any point of X. Since F is maximal, by Corollary 2, F does not - -accumulates to any point of X.
This implies that for every x 2 X, there exists an -open set Vxand an Fi(x)2 F
such that Fi(x)\ Vx = . The family fVx : x 2 Xg is an -open cover of X and
by hypothesis, there exists a …nite number of points x1; x2; :::; xn of X such that
X = [fV(xj): j = 1; 2; :::; ng. Since F is a …lterbase on X, there exists an F02 F
such that F0 \fFi(xj): j = 1; 2; :::; ng. Hence F0\ V(xj) = for j = 1; 2; :::; n.
Which implies that F0\ f[V(xj) : j = 1; 2; :::; ng = F0\ X = . Therefore, we
obtain F0= . Contracting the fact that F06= .
(2) ) (3): Let F be any …lterbase on X. Then, there exists a maximal …lterbase F0 such that F F0. By hypothesis, F0 - -converges to some point x 2 X. For
every F 2 F and every -open set V containing x, there exists an F0 2 F0 such
that F0 V , hence 6= F0\ F V \ F . This shows that F - -accumulates at
x.
(3) ) (1): Let U = fUi: i 2 Ig be an -open cover of X such that X 6= [ni=1Ui.
Let B denote the set of all sets of the form Tni=1(Ui)c. Since
T
(Ui)c 6= , B is
…lterbase in X and so by our assumption, it - -accumulates to some point x 2 X. But then x does belong to some U 2 U and so (U )c 2 B yields a contradiction
(U )c\ U 6= . This completes the proof.
Proposition 3. Let X be an - -compact space. Then, for every regular open cover fFi: i 2 Ig of X, there exists a …nite subset I0 of I such that X =SfFi : i 2 I0g.
Proof. Let fFi : i 2 Ig be any regular open cover of X. Since, Fi 2 O(X) for
each i 2 I, then the family fFi: i 2 Ig forms -open of X, since X is - -compact
space, there exists a …nite subset I0of I such that X =
S
fFi : i 2 I0g.
Proposition 4. Let X be an - -compact space. Then, for every family fX n Vi:
i 2 Ig of regular closed subsets of X such that TfX n Vi: i 2 Ig = , there exists
a …nite subset I0 of I such that TfX n Vi : i 2 Ig = .
Proof. let fX n Vi : i 2 Ig be a family of regular closed such that
T
fX n Vi : i 2
Ig = . So fVi : i 2 Ig is a family of regular open and X =
S
fVi : i 2 Ig, by
Proposition 3, there exists a …nite subset I0 of I such that X =
S
fVi : i 2 I0g
implies that =TfX n Vi : i 2 I0g.
De…nition 8. A subset A of a topological space (X; ) is said to be - -compact of X if for every -open cover fVi: i 2 Ig of A, there exists a …nite subset I0 of I
such that A SfVi : i 2 I0g.
Theorem 6. Let (X; ) be a topological space, K be a subset of X and be operation on O(X) which is -stable with respect to K. If X is - -compact and K is -closed, then K is - K-compact.
Proof. Let U = fU g 2I be an -open cover of K by jK. Lets U be the family of
all -open sets such that for each V 2 U , V \K 2 U. Since XnK is -open, we can take an -open cover of X n K say W = fWx2 O(X) : Wx X n K; x 2 X n Kg.
we have two …nite subcollections fV1; :::; Vng U and fW1; :::; Wmg W such
that X = f[n
i=1Vi g [ f[mj=1Wjg. Then K = f[ni=1Vi \ Kg [ f[mj=1Wj \ Kg =
[n
i=1(Vi\K) K = [ni=1(Ui) K, since Wj \K = for j = 1; 2; :::; m and is -stable
with respect to K. Therefore K is - K-compact.
Theorem 7. Let (X; ) be a topological space and K be a subset of X. Let be an operation on O(X) and -stable with respect to K. Then K is - -compact if and only if K is - K-compact.
Proof. Suppose that K X is - -compact and let C be an -open cover of K by jK. Then the set C of all G 2 O(X) with G \ K 2 C is an -open cover of K, and hence we can …nd a subfamily fG1; G2; :::; Gng of C such that K [ni=1Gi.
Therefore we have
K = ([ni=1Gi) \ K = [ni=1(Gi \ K) = [ni=1(Gi\ K) K.
Conversely, suppose that K is - K-compact. If C is an -open cover of K, then fG \ K : G 2 O(X)g jK is a cover of K, and so there exists a …nite subfamily fG1; G2; :::; Gng of C such that
K = [n
i=1(Gi\ K) K = [ni=1(Gi \ K) [ni=1Gi.
Theorem 8. Let K be a subset of X, : O(X) ! P (X) and K : jK ! P (K) be operations satisfying the following properties, (V \ K) K V \ K for any
-open set V of X such that V \ K 6= . If K is - K-compact in (K; jK), then K
is - -compact.
Proof. Let C be an -open cover of K. Then fG \ K : G 2 Cg jK is a cover of K and so there exists a …nite subfamily fG1; G2; :::; Gng of C such that
K = [n
i=1(Gi\ K) K [n
i=1Gi \ K [ni=1Gi. Therefore, K is - -compact.
Theorem 9. A space X is - -compact if and only if every proper -closed subset of X is - -compact.
Proof. Let F be any proper -closed subset of X. Let fVi : i 2 Ig be an -open
cover of F . Since F is -closed set, then X n F is -open set. So the family fVi: i 2 Ig [ X n F is an -open cover of X. Since X is - -compact, there exists
a …nite subset I0 of I such that X = SfVi : i 2 I0g [ (X n F ) . Therefore, we
obtain F SfVi : i 2 I0g [ (X n F ) . Hence F is - -compact of X.
Conversely, let fVi: i 2 Ig be an -open cover of X. Suppose that X 6= Vi06=
for every i0 2 I. Then X n Vi0 is a proper -closed subset of X. Therefore, by
hypothesis, there exists a …nite subset I0 of I such that X n Vi0 [fVi : i 2 I0g.
Therefore, we obtain X = SfVi : i 2 I0g [ Vi0. Which shows that X is
-compact.
Theorem 10. Let A and B be subsets of a space X such that A \ B 6= . If A is - -compact subset of X and B is -closed set, then A \ B is - -compact subset of X.
Proof. Let fVi : i 2 Ig be any -open cover of A \ B. Since B is -closed set, then
X n B is -open. So the family fVi: i 2 Ig [ X n B is an -open cover of A. Since
A is - -compact subset of X, then there exists a …nite subset I0 of I such that
A fVi : i 2 I0g[(X nB) . Therefore, we obtain A\B fVi : i 2 I0g[(X nB) .
Hence A \ B is - -compact subset of X.
Corollary 3. Let A be an - -compact subset of X. If B is an -closed set of X and B A, then B is - -compact subset of X.
Theorem 11. Let A be any subset of a topological space (X; ) such that A and X n A are - -compact subsets of X, then X is - -compact.
Proof. Let fVi : i 2 Ig be any -open cover of X = A [ X n A. Then fVi: i 2 Ig
is an -open cover of A and X n A. Therefore, there exists …nite subfamilies I0
and I1 of I such that A SfVi : i 2 I0g and X n A SfVi : i 2 I1g. Thus,
X = A [ X n A SfVi : i 2 I0[ I1g. This completes the proof.
Corollary 4. The …nite union of - -compact subsets of X is - -compact. Proof. Straightforward.
Theorem 12. Let B be - -compact subset of X and G be -open subset of a space X such that G B. Then B n G is - -compact subset of X.
Proof. Obvious.
Theorem 13. Let (X; ) be a topological space and be -regular operation on O(X). If X is - -T2 and K X is - -compact, then K is -closed.
Proof. We need to prove that X nK is -open. So let x02 X nK. For each y 2 K,
there exists -open sets Uy and Vy such that x0 2 Uy, y 2 Vy and Uy \ Vy = .
In this way we construct an -open cover U = fVy : y 2 Kg of K. Since K is
-compact, there exists a …nite collection fVy1; :::; Vyng of U such that K [ n i=1Vyi.
Let U = \n
i=1Uyi. We can see that U is an -open set containing x0, but it
does not have to happen that U X n K. Here we need the -regularity of to achieve our purpose. Since Uy1; :::; Uyn are -open sets containing x0, then
using the -regularity of there exists an -open set W containing x0, such that
W W X n K. This implies that X n K is -open, and hence K is -closed.
Theorem 14. Let (X; ) be a topological space and an -additive operation on O(X). If Y X is - -compact, x 2 X n Y and (X; ) is - -T2, then there exist
-open sets U and V with x 2 U, Y V and U \ V = .
Proof. For each y 2 Y , let Vyand Vxy be -open sets such that Vy \ Vxy = , with
y 2 Vyand x 2 Vxy. The collection V = fVy: y 2 Y g is an -open cover of Y . Now,
since Y is - -compact, there exists a …nite subcollection fVy1; :::; Vyng of V such
that Y [n
i=1Vyi. Let U = \ n
i=1Vxyi and V = [ni=1Vyi. Since U V yi
i 2 f1; 2; :::; ng, then U \ Vyi= for every i 2 f1; 2; :::; ng. Then U \ V = as
is an -additive operation on O(X) and Y V .
Theorem 15. Let (X; ) be a topological space and an -additive operation on O(X). Suppose thatTki=1Ki (Ti=1k Ki) for any collection fK1; K2; :::; Kkg
O(X). If A and B are disjoint - -compact subsets of X and (X; ) is - -T2,
then there exist disjoint -open sets U and V such that A U and B V . References
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Current address : Alias B. Khalaf: Department of Mathematics, Faculty of Science, University of Duhok, Iraq.
E-mail address : aliasbkhalaf@gmail.com
Current address : Hariwan Z. Ibrahim: Department of Mathematics, Faculty of Science, Uni-versity of Zakho, Iraq
