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0Journal of the Egyptian Mathematical Society (2013) 21, 63-67
Egyptian Mathematical Society Journal of the Egyptian Mathematical Society
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ORIGINAL ARTICLE
On b-connectedness and b-disconnectedness and their applications
A.A. El-Atik a *, H.M. Abu Donia b, A.S. Salama a
a Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt b Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt
Received 7 March 2012; revised 21 July 2012; accepted 1 September 2012 Available online 2 November 2012
KEYWORDS
b-Separated sets; b-Connected sets
Abstract In this paper, by using b-open ( = y-open) sets we study the concept of b-separated sets. With this concept we study the notion of b-connected sets and strongly b-connected sets. We give some properties of such concepts with some b-separation axioms and compact spaces. Finally, we construct a new topological space on a connected graph.
2000 AMS SUBJECT CLASSIFICATION: 54B05, 54B10, 54C10, 54D18, 90D42
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1. Introduction
Connectedness [4] is a well-known notion in topology. Numerous authors studied connectedness. In [5], P-spaces and extremal disconnectedness are studied. Connectedness in [19-21] are used to expand some topological spaces. In [27], authors proved that neither first countable nor C~ech-complete spaces are maximal Tychonoff connected. Many other topologists defined and studied connectedness in bitopological spaces [8,26]. It is important to study some types of connectedness in digital spaces. A point with integer coordinates is called a digital point. The problem of finding a topology for the digital plane
Corresponding author. E-mail address: aelatik55@yahoo.com (A.A. El-Atik). Peer review under responsibility of Egyptian Mathematical Society.
and the digital 3-space is of importance in image processing and more generally in all situations where spatial relations are modeled on a computer. In all these applications it is essential to have a data structure on the computer which shares as many as possible features with the real topological situation.
Connectedness and compactness are powerful tools in topology but they have many dissimilar properties. The concept of Hausdorff spaces is almost an integral part of compactness. Investigations into the properties of cut points of topological spaces which are connected, compact and Hausdorff date back to the 1920s. Connectedness together with compactness with the assumption of Hausdorff has been studied in [28] from the view point of cut points. In [22], authors studied some types of connected topological spaces. The class of b-open sets in the sense of Andrijevic [3] was discussed by El-Atik [17] under the name of y-open sets. Since then these concepts have used to define and investigate many topological properties. The aim of this paper is to study y-connectedness. Also digital spaces are examined in the context of these new
1110-256X © 2012 Egyptian Mathematical Society. Production and hosting by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/jjoems.2012.09.003
concepts. However, our main interest shall be digital spaces that are also topological spaces.
2. Preliminaries
In [3], Andrijevic introduced a new class of generalized open sets called b-open ( = y-open [17]) sets into the field of topology. This class is a subset of the class of semi-preopen sets [2], i.e. a subset of a topological space which is contained in the closure of the interior of its closure. Also the class of b-open sets is a superset of the class of semi-open sets [23], i.e. a set which is contained in the closure of its interior, and the class of locally dense sets [7] or preopen sets [24], i.e. a set which is contained in the interior of its closure. Andrijevic studied several fundamental and interesting properties of b-open sets. In [6,9-11] authors discussed some applications of b-open sets.
Throughout the present paper, the space (X, s) and (Y, r) always mean topological spaces on which no separation axioms are assumed unless explicitly stated. A subset A is said to be semi-open [23] (resp. b-open [1], preopen [24], a-open [25]) if A c Cl(Int(A)) (resp. A c Cl(Int(Cl(A))), A c Int(Cl(A))), A c Int(Cl(Int(A))), where Cl(A) and Int(A) denote the closure and the interior of A in (X, s), respectively. The complement of a semiopen (resp. b-open, preopen, a-open) set is said to be semi-closed (resp. b-closed, preclosed,a -closed). We denote the collection of all semiopen (resp. b-open, preopen, a-open) sets by SO(X) (resp. bO(X), PO(X), aO(X)). We set SO(X, x) = {U: x 2 U 2 SO(X)}, bO(X, x) = { U: x 2 U 2 bO(X)}, PO(X,-x) = {U: x 2 U 2 PO(X)} and aO(X,x) = {U: x 2 U 2 aO(X)}.
Let A c X, then A is said to be b-open [3] if A c Cl(In-t(A)) U Int(Cl(A)). The complement X — A of a b- open set A is called b-closed and the b-closure of a set A, denoted by bCl(A), is the intersection of all b-closed sets containing A. bCl(A) is the smallest b-closed set containing A. The b-interior of a set A denoted by bInt(A), is the union of all b-open sets contained in A. bInt(A) is the largest b-open set contained in A. The family of all b-open (resp. b-closed) sets in a space X will be denoted by BO(X) (resp. BC(X)).
Proposition 2.1 [3]. (i) The union of any family of b-open sets is a b-open set. (ii) The intersection of an open and a b-open set is a b-open set.
Lemma 2.2. The b-closure of a subset A of X, denoted by bCl(A), is the set of all x 2 X such that O n A „ / for every O 2 BO(X,x), where BO(X,x) = {U: x 2 U 2 BO(X,s)}.
Definition 2.3 12. The b-boundary of a set A of a space X is defined by b-bd(A) = bCl(A) n bCl(X — A).
Definition 2.4 [13]. A space X is said to be b-connected if X cannot be expressed as the union of two disjoint nonempty b-open sets of X.
Lemma 2.5 17. Let Abe a subset of a topological space X. Then A 2 BO(X) if and only ifbCl(A) is b-clopen in X (i.e., b-open and b-closed).
Definition 2.6 17. A subset N c X is called a b-neighborhood (breifly b-nbd) of a point x 2 X if there exists a b-open set U c N such that x 2 U c N.
3. b-Separateness and b-connectedness
Definition 3.1 13. Two subsets A and B in a space X are said to be b-separated if and only if A n bCl(B) = / and bCl(A) n B = /.
From the fact that bCl(A) c Cl(A), for every subset A of X, every separated set is b-separated. But the converse may not be true as shown in the following example.
Example 3.2. Let X = {a, b, c, d} with a topology s = {X, /, {a}, {b}, {a,b}}. The subsets {a}, {c,d} are b-separated but not separated.
Remark 3.3. Each two b-separated sets are always disjoint, since A n B c A n bCl(B) = /. The converse may not be true in general.
Example 3.4. Let X = {a, b, c, d} with a topology s = {X, /,
{a}, {b, c}, {a,b,c}}. BO(X) = {X, /, {a}, {b}, {c}, {a,b}, {a, c}, {b, c}, {a,b, c}, {a,b,d}, {a, c,d}, {b,c,d}} The subsets {a, c}, {b, d} are disjoint but not b-separated.
Theorem 3.5. Let A and B be nonempty sets in a space X. The following statements hold:
(i) If A and B are b-separated and A1 c A and B1 c B, then A1 and B1 are so.
(ii) If A n B = / such that each of A and B are both b-closed (b-open), then A and B are b-separated.
(iii) If each of A and B are both b-closed (b-open) and if H = A n (X — B) and G = B n (X — A), then H and G are b-separated.
Proof.
(i) Since At c A, then bCl(A1) c bCl(A). Then B n bCl(A) = / implies B! n bCl(A) = / and B! n bCl(A1) = /. Similarly A! n bCl(B1) = /. Hence A! and B1 are b-separated.
(ii) Since A = bCl(A) and B = bCl(B) and A n B = /, then bCl(A) n B = / and bCl(B) n A = /. Hence A and B are b-separated. If A and B are b-open, then their complements are b-closed.
(iii) If A and B are b-open, then X — A and X — B are b-closed. Since H c X — B, bCl(H) c bCl(X — B) = X — B and so bCl(H) n B = /. Thus G n bCl(H) = /. Similarly, H n bCl(G) = /. Hence H and G are bseparated. □
Theorem 3.6. The sets A and B of a space X are b-separated if and only if there exist U and V in BO(X) such that A c U, B c V and A n V = /, B n U = /.
Proof. Let A and B be b-separated sets. Set V = X — bCl(A) and U = X — bCl(B). Then U, V 2 BO(X) such that A c U, B c V and A n V = /, B n U = /. On the other hand, let U, V 2 BO(X) such that A c U, B c V and A n V = /, B n U = /. Since X — V and X — U are b-closed, then
bCl(A) с X - Vс X - B and bCl(B) с X - Uс X - A. Thus bCl(A) П B = / and bCl(B) n A = /. □
Definition 3.7 17. A point x 2 X is called a b-limit point of a set A с X if every b-open set U с X containing x contains a point of A other than x.
Theorem 3.8. Let A and B be nonempty disjoint subsets of a space X and E = A U B. Then A and B are b-separated if and only if each of A and B is b-closed (b-open)in E.
Proof. Let A and B are b-separated sets. By Definition 3.1, A contains no b-limit points of B. Then B contains all b-limit points of B which are in A U B and B is b-closed in A U B. Therefore B is b-closed in E. Similarly A is b-closed in E. □
Definition 3.9 13. A subset S of a space X is said to be b-connected relative to X if there is not exist two b-separated subsets A and B relative to X and S = A U B. Otherwise, S is said to be b-disconnected.
By Definition 3.9, one can show that each b-connected set is connected. The converse may not be true in general as shown in [13] and in the below examples. In other words, each disconnected is b-disconnected.
Example 3.10. A space X = {a, b, c} with Sierpinski topology is connected but not b-connected.
Example 3.11. Any space with indiscrete topology is connected but not b-connected since b-open sets establish a discrete topology.
Example 3.12. Let X = {a, b, c, d} with a topology s = {X, /, {a}, {a,b}}. The subset {a,b, c} is connected but not b-connected.
Theorem 3.13. Let A and B be subsets in a space X such that A с B с bCl(A). If A is b-connected, then B is b-connected.
Proof. If B is b-disconnected, then there exist two b-separated subsets U and V relative to X such that B = U U V. Then either A с U or A с V. Without loss of generality, let A с U. As A с Uс B, bClB(A) с bClB(U) с bCl(U). Also bClB(A) = B n bCl(A) = B □ bCl(U). This implies to B = bCl(U). So U and V are not b-separated and B is b-connected. □
Definition 3.14. A space X is locally b-connected at a point p if every b-nbd of p contains a b-connected b-nbd of p. X is said to be locally b-connected if it is locally b-connected at each of its points.
The following lemma is due to El-Atik [17].
Lemma 3.15. Let A and X0 be subsets of a space X. If A 2 BO(X) and X0 2 a.O(X), then A n X0 2 BO(X0).
Lemma 3.16. Every a-open subspace X0 of a locally b-connected space is locally b-connected in X0.
Proof. This is an immediate consequence of Definitions 3.14, 2.6 and Lemma 3.15. □
Theorem 3.17. Any space X is locally b-connected if the components of every a-open subspace U of X are b-open in U.
Proof. Let X be a locally b-connected space and let U be an a-open subspace of X. By Lemma 3.16, U is a locally b-connected space in U. Also, the components of U say, St are a-open sets of U. Since U is a-open, then by Lemma 3.15,
Ai = Ai П U 2 BO(U). □
Theorem 3.18. If E is b-connected, then bCl(E) is b-connected.
Proof. By contradiction, suppose that bCl(E) is b-discon-nected. Then there are two nonempty b-separated sets G and H in X such that bcl(E) = G U H. Since E = (G П E) U (H П E) and bCl(G П E) с bCl(G) and bCl(H П E) с bcl(H) and G П H = /, then (bCl(G П E)) П H = /. Hence (bCl(G П E)) П (H П E) = /. Similarly (bcl(H П E)) П (G П E) = /. Therefore E is b-disconnected. □
l.emma 3.19. Let A ç B U C such that A be a nonempty b-connected set in a space X and B, C are b-separated. Then only one of the following conditions holds:
(i) A ç B and A П C = /.
(ii) A ç C and A П B = /.
Proof. Since A П C = /, then A ç B. Also, if A П B = /, then A ç C. Since A ç B П C, then both A П B = / and A П C = / cannot hold simultaneously. Similarly, suppose that A П B „ / and A П C „ /, then, by Theorem 3.5(i), A П B and A П C are b-separated such that A = (A П B) U (A П C) which contradicts with the b-connect-edness of A. Hence one of the conditions (i) and (ii) must be hold. □
Definition 3.20 13, 17. A functionf: X fi Y is said to be:
(i) b-continuous if the inverse image of each open set in Y is b-open X.
(ii) b-open if the image of each open set in X is b-open Y.
(iii) b-closed if the image of each closed set in X is b-closed Y.
Lemma 3.21 17. Letf: X fi Y be a b-continuous function. Then ba(r'(B)) c f-1(Cl(B)),for each B c Y.
Theorem 3.22. For a b-continuous function f: X fi Y, if K is b-connected in X, then f(K) is connected in Y.
Proof. Suppose that f(K) is disconnected in Y. There exist two separated sets P and Q of Y such that f(K) = P U Q. Set A = K nf"1(P) and B = K nf"1(Q). Since f(K) n P „ /, then K n fl(P) „ / and so A „ /. Similarly B „ /. Since P n Q = / , then A n B = K n/^1(P n Q) = / and so A n B = /. Since f is b-continuous, then by Lemma 3.21,
bCl(Tlm crl(Cl(Q)) and B cr\Q), then
bCl(B) cf—1(Cl(Q)). Since P n Cl(Q) = /, then
A nr\a(Q)) cf—1(P) n r\Cl(Q)) = / and then A n bCl(B) = /. Thus A and B are b-separated. □
Corollary 3.23. For a b-continuous function f: X fi Y, if K is disconnected in X, then f(K) is b-disconnected in Y.
Proof. Obvious. □
Theorem 3.24. For a bijective b-closed f: X fi Y, if K is b-con-nected Y, then f—1(K) is connected X.
Proof. The proof is similar to that of Theorem 3.22. Thus we omit it. □
To avoid the confusion for definitions of b-open and b-closed functions in [17,16] and [6], we set M-b-open and M-b-closed functions instead of b-open and b-closed functions.
Definition 3.25. A function f: X fi Y is said to be:
(i) b-Irresolute [6] if for each point x 2 X and each b-open set V of Y containing f(x), there exists a b-open set U of X containing x such that f(U) c V.
(ii) b-Irresolute [17] iff—1(V) 2 BO(X) for every V2 BO(Y).
(iii) M-b-open [16] iff(V) 2 BO(Y) for every V2 BO(X).
(iv) M-b-closed [14] if f(V) c BC(Y) for every V 2 BC(X).
(v) Strongly b-irresolute [17] if f—1(V) 2 BO(X) for every open set V in Y.
(vi) Strongly M-b-open [17] if f(V) 2 BO(Y) for every open set V in X.
(vii) Strongly M-b-closed [17] if f(V) 2 BC(Y) for every closed set V in X.
Lemma 3.26. [17] A function f: X fi Y is a b-irresolute if and only if bcl(f—1(B)) cf—1(bcl(B)) cf—1 (cl(B)), for each Bc Y.
Theorem 3.27. Let f: (X, s) fi (Y, r) be a b-irresolute function. If K is b-connected in X, then f(K) is b-connected in Y.
Proof. By using Definition 3.25 and Lemma 3.26, it is direct consequence of Theorem 3.22. □
4. Strongly b-connectedness in compact spaces
Definition 4.1. A space X is strongly b-connected if and only if it is not a disjoint union of countably many but more than one b-closed set i.e. if Et are nonempty disjoint closed sets of X, then X„ Ei U E2 U E2 U ■ ■ ■. otherwise X is said to be strongly b-disconnected
Note the similarity between Definition 4.1 and that of b-connectedness. If X is b-connected, and E1 and E2 are any two nonempty disjoint closed sets of X, then X„ E1 U E2.
Lemma 4.2. For any surjective b-irresolute function f: X fi Y. The image f(X) is strongly b-connected if X is strongly b-connected.
Proof. Suppose f(X) is strongly й-disconnected, by Definition 4.1 it is a disjoint union of countably many but more than one й-closed sets. Since fis b-irresolute, then the inverse image of й-closed sets are still b-closed, Xis also a disjoint union of b-closed sets. Therefore, f(X) is strongly b-connected. □
Theorem 4.3. A space X is strongly b-connected if there exists a constant surjective b-irresolute function f: X fi D, where D denote to a discrete space of X.
Proof. Let X be strongly b-connected and f: X fi D be a surjective b-irresolute function, then by Lemma 4.2, f(X) is strongly b-connected. The only strongly b-connected subset of D are the one-point spaces. Hence f is constant. Conversely, suppose X is a disjoint union of countably many but more than one b-closed sets, X = uiEi. Then define f: X fi D by taking f(x) = i whenever x 2 E;-. This f is a surjective b-irresolute and not constant. So X is strongly b-connected. □
Strongly b-connectedness is a stronger notion of b-connect-edness. In other words, given a b-connected space, we can make it strongly b-connected by adding some conditions. But what conditions should be added is the difficulty. Our starting point is b-connected spaces, thus a b-continuum may be useful. The concept of a b-continuum is defined on a b-connected set.
Definition 4.4. A compact b-connected set is called a b-continuum.
Definition 4.5. A space X is called:
(i) bT1 [15] if for each x, y 2 X, x „ y, there exist two disjoint b-open sets U and V such that x 2 U, y R U and
x R V, y 2 V.
(ii) bT2 ([15,18]) if for each x, y 2 X, x „ y, there exist two disjoint b-open sets U and V such that x 2 U, y 2 V and U П V = /.
(iii) b-normal (y-normal [14]) for any pair of disjoint b-closed sets F1 and F2, there exist disjoint b -open sets U and V such that F1 с U and F2 с V such that U П V = /.
Lemma 4.6. If A is any b-continuum in a bT2 space X and B is any b-open set such that A n B „ / „ A n (X — B), then every component of (A n bCl(B)) n b — bd(B) „ /.
Proof. It is obvious by Definitions 2.3, 4.4 and 4.5. □
Theorem 4.7. Let X be a compact bT2-space. Then X is b-con-nected if and only if X is strongly b-connected.
Proof. It is clear that if X is strongly b-connected, then X is b-connected. Now, suppose that Xis a compact bT2 b-connected space and it is strongly b-disconnected, then X is a union of a countably many but more than one disjoint b-closed sets. Then X = UK, where Kt are b-closed disjoint sets. Since a compact bT2-space is b-normal, then X, by Definition 4.5, is a b-normal space. So there exist a b-open sets U such that K2 c U and bCl(U) n K1 = /. Let X1 be a component of bCl(U) which intersects K2. Then X1 is compact and b-connected. Now by Lemma 4.6, X1 n b — bd(U) „ / i.e. X1 contains a point
p 2 b — bd(U) such that p R U and p R K1. Hence X! П K „ / for some i > 2. Let Kn2 be the first Ki for i > 2 which intersects X!, and let V be a b-open set satisfying Kn2 с V, and bCl(V) П K2 = /. Then let X2 be a component of Xl П bCl(V) which contains a point of Kn2 . Again we have X2 П b — bd(V) „ /, and X2 contains some pointp 2 b — bd(V) such that p R V, p R K1 U K2. Hence X2 П K „ / for some i > n2, and X2 П K = / for i < n2. Let Kn3 be the first K for i > n2, which intersects X2, then by methods similar to the above we can find a compact b-connected X3 such that X3 с X2 с X1, and X3 intersects some K with i > n3 but X3 П K = /for i < n3. In this manner, we obtain a sequence of subcontinua of X: X1 X2 X3 ■■■, such that for each j, Xj П K = / for i < nj and nj fi i as j fi i. we know that niXi „ /. Also, (niX) П K = / for all j, so that (ntX) П (и,-Kj) = / or (nX) П X = /. But (ntXt) с X, which contradicts the fact that niXi „ /. Therefore X strongly b-connected. □
Theorem 4.8. Let X be a locally compact bT2-space. If X is locally b-connected, then X is locally strongly b-connected.
Proof. Let O be a b-open b-nbd of a point x 2 X. Then there exists a compact b-nbd V of x lying inside O. Let C be a b-connected component of V containing x. Since V is a b-nbd of x and X is locally b-connected, C is a b-nbd of x. Since C is b-closed in V and V is compact, then C is compact. So C is a compact b-connected b-nbd of x lying inside O. By Theorem 4.7, C is strongly b-connected. □
Theorem 4.9. Let X be a locally compact bT2-space. If X is locally b-connected and b-connected, then X is strongly b-connected.
Proof. This follows from Theorems 4.7 and 4.8. □
Lemma 4.10 (15,18). For a space X the following statements are equivalent:
(i) X is a bT1-space.
(ii) For any point x 2 X, the singleton set {x} is b-closed.
Corollary 4.11. A strongly b-connected b Tj-space having more than one point is uncountable.
Proof. By Lemma 4.10, a one-point set in a b Trspace is b-closed. Thus by Definition 4.1, a b T1-space cannot have countably many but more than one point. □
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