- Email: [email protected]
On locally δ-generalized closed sets and LδGC-continuous functions
Chaos, Solitons and Fractals 19 (2004) 995–1002 www.elsevier.com/locate/chaos
On locally d-generalized closed sets and LdGC-continuous functions Jin Keun Park a, Jin Han Park a
a,*
, Bu Young Lee
b
Division of Mathematical Sciences, Pukyong National University, 599-1, Daeyeon 3-Dong, Nam-Gu, Pusan 608-737, South Korea b Departments of Mathematics, Dong-A University, 840, Hadan 2-Dong, Saha-Gu, Pusan 604-714, South Korea Accepted 28 April 2003
Abstract In this paper we introduce ldgc-sets, ldgc -sets and ldgc -sets and different notions of generalizations of continuous functions in a topological space and study some of their relationship and their properties. Possible application to computer graphics and quantum physics are touched upon. Ó 2003 Elsevier Ltd. All rights reserved.
1. Introduction In 1968, Velicko [17] introduced d-open sets in order to investigate the characterization of H-closed spaces and showed that sd , i.e. the collection of all d-open sets, is the topology on X such that sd s and, in particular, sd equal to the semi-regularization topology ss . The approach of the concept of semi-regularization from a different perspective via the concept of generalized closedness was done by Dontchev and Ganster [3]. They consider sets whose closure in the semi-regularization are contained in every superset which is open in the original topology. They called these sets dgeneralized closed (briefly, dg-closed) and studied their basic properties. The notion of a locally closed set in a topological space was implicitly introduced by Kuratowski and Sierpienski [11]. According to Bourbaki [2] a subset of a topological space X is locally closed in X if it is the intersection of an open set and a closed set in X . In 1989, Ganster and Reilly [6] continued the study of locally closed sets and also introduced the concept of LC-continuous functions to find a decomposition of continuous functions. Thereafter, intensive research on the field of locally closed sets and LCcontinuous functions has been done as theory developed by Balachandran et al. [1], Ganster et al. [6], Nasef [14] and Park and Park [16]. Several topological spaces that fail to be T1 are very often of significant importance in digital topology, i.e. in the study of the geometric and topological properties of digital images [9,10]. Such is the case with the major building block of the digital n-space, the digital line or the so called Khalimsky line [7–10]. This is the set of the integers, Z, equipped with the topology K, generated by GK ¼ ff2n 1; 2n; 2n þ 1g : n 2 Zg. The Khalimsky line, a space which is widely used in the applications of point-set topology in computer graphics, is T3=4 but not T1 [3]. The purpose of this paper is to introduce three new classes of sets called ldgc-sets, ldgc -sets and ldgc -sets, which are contained in the class of glc-sets and contain the class of ldc-sets by using the notions of d-open and d-closed sets due to Velicko [17] and dg-open and dg-closed sets due to Dontchev and Ganster [3]. Also, we introduce some different classes of continuity and irresoluteness and study some of their properties.
*
Corresponding author. E-mail address: [email protected] (J.H. Park).
0960-0779/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0960-0779(03)00194-2
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2. Preliminaries All topological spaces considered in this paper lack any separation axioms unless explicitly stated. The topological space is denoted by ðX ; sÞ will be replaced by X if there is no chance for confusion. For a subset A X , the closure and the interior of A in X are denoted by clðAÞ and intðAÞ, respectively. The d-interior [17] of a subset A of X is the union of all regular open sets of X contained in A and is denoted by d-intðAÞ. A subset A is called d-open [17] if A ¼ d-intðAÞ, i.e. a set is d-open, if it is the union of regular open sets. The complement of a d-open set is called d-closed. Alternatively, a set A ðX ; sÞ is called d-closed [17] if A ¼ d-clðAÞ, where d-clðAÞ ¼ fx 2 X : intðClðU ÞÞ \ A 6¼ /; for every open set U containing xg. The family of all d-open sets form a topology on X and is denoted sd . Recall the following definitions which will be used in sequel. Definition 2.1. A subset A of a space X is called: (a) g-closed [13] if clðAÞ G whenever A G and G is open in X ; (b) dg-closed [3] if d-clðAÞ G whenever A G and G is open in X ; (c) g-open (resp. dg-open) if the complement of A is g-closed (resp. dg-closed).
Definition 2.2. A subset A of ðX ; sÞ is called: (a) (b) (c) (d) (e)
locally closed set [5] if A ¼ G \ F where G 2 s and F is closed in ðX ; sÞ; locally d-closed set [15] (briefly, ldc-set) if A ¼ G \ F where G is d-open and F is d-closed in ðX ; sÞ; generalized locally closed set [1] (briefly, glc-set) if A ¼ G \ F where G is g-open in ðX ; sÞ and F is g-closed in ðX ; sÞ; glc -set [1] if there exist a g-open set G and a closed set F of ðX ; sÞ such that A ¼ G \ F ; glc -set [1] if there exist an open set G and a g-closed set F of ðX ; sÞ such that A ¼ G \ F .
The collection of all locally d-closed sets (resp. glc-sets, glc -sets, glc -sets) of ðX ; sÞ will be denoted by LdCðX ; sÞ (resp. GLCðX ; sÞ, GLC ðX ; sÞ, GLC ðX ; sÞ.) Definition 2.3. A function f : ðX ; sÞ ! ðY ; rÞ is called: (a) (b) (c) (d)
LdC-continuous [14] if f 1 ðV Þ 2 LdCðX ; sÞ for each V 2 r; GLC -continuous [1] if f 1 ðV Þ 2 GLC ðX ; sÞ for each V 2 r; GLC -continuous [1] if f 1 ðV Þ 2 GLC ðX ; sÞ for each V 2 r; GLC-continuous [1] if f 1 ðV Þ 2 GLCðX ; sÞ for each V 2 r.
Lemma 2.4 [3] (a) Finite union of dg-closed sets is a dg-closed set. (b) The intersection of a dg-closed set and a d-closed set is dg-closed.
3. Locally d-generalized closed sets We begin with the following definitions as a generalization of ldc-sets. Definition 3.1. A subset A of ðX ; sÞ is called: (a) locally d-generalized closed set (briefly, ldgc-set) if there exist a dg-open set G and a dg-closed set F of ðX ; sÞ such that A ¼ G \ F ; (b) ldgc -set if there exist a dg-open set G and a d-closed set F of ðX ; sÞ such that A ¼ G \ F ; (c) ldgc -set if there exist a d-open set G and a dg-closed set F of ðX ; sÞ such that A ¼ G \ F . The collection of all ldgc-sets (resp. ldgc -sets, ldgc -sets) of ðX ; sÞ will be denoted by LdGCðX ; sÞ (resp. LdGC ðX ; sÞ, LdGC ðX ; sÞ).
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Remark 3.2. From the definitions above and Definition 2.2, we have the following implications: ldgc -set %
! &
ldc-set
glc -set ldgc-set
& ldgc -set
% !
& ! %
glc-set
glc -set
In the remark above, the relationships cannot be reversible as the following examples illustrate. Example 3.3. Let X ¼ fa; b; cg and s ¼ fX ; /; fag; fcg; fa; cg; fb; cgg. (a) fcg 2 LdGC ðX ; sÞ but fcg 62 LdCðX ; sÞ. (b) fa; bg 2 LdGC ðX ; sÞ but fa; bg 62 LdCðX ; sÞ. (c) fa; bg 2 LdGCðX ; sÞ but fa; bg 62 LdGC ðX ; sÞ. (d) fcg 2 LdGCðX ; sÞ but fcg 62 LdGC ðX ; sÞ. (e) fa; cg 2 GLC ðX ; sÞ but fa; cg 62 LdGC ðX ; sÞ. (f) fcg 2 GLC ðX ; sÞ but fcg 62 LdGC ðX ; sÞ. Example 3.4. Let X ¼ fa; b; c; dg and s ¼ fX ; /; fag; fbg; fa; bg; fa; b; cg; fa; b; dgg. Then fcg 2 GLCðX ; sÞ but fcg 62 LdGCðX ; sÞ. Recall that a topological space ðX ; sÞ is called semi-regular [12] if s ¼ ss and a topological space ðX ; sÞ is called T3=4 [3] if every dg-closed set is d-closed in ðX ; sÞ. Remark 3.5 (a) If a space ðX ; sÞ is semi-regular, then LdGC ðX ; sÞ ¼ LdGC ðX ; sÞ ¼ GLCðX ; sÞ: (b) If a space ðX ; sÞ is T3=4 , then LdGCðX ; sÞ ¼ LdCðX ; sÞ: Now, we obtain a characterization for ldgc -sets as follows: Theorem 3.6. For a subset A of ðX ; sÞ, the following are equivalent: (a) (b) (c) (d)
A 2 LdGC ðX ; sÞ. A ¼ G \ d-clðAÞ for some dg-open set G. d-clðAÞ n A is dg-closed. A [ ðX n d-clðAÞÞ is dg-open.
Proof (a)!(b): Let A 2 LdGC ðX ; sÞ. Then A ¼ G \ F , where G is dg-open and F is d-closed. A F implies d-clðAÞ F and so A ¼ G \ F G \ d-clðAÞ. And A G and A d-clðAÞ implies A G \ d-clðAÞ. Hence A ¼ G \ d-clðAÞ. (b)!(a): Since G is dg-open and d-clðAÞ is d-closed, G \ d-clðAÞ 2 LdGC ðX ; sÞ by Definition 3.1(b). (b)!(c): d-clðAÞ n A ¼ d-clðAÞ \ ðX n GÞ, which is dg-closed by Lemma 2.4(b). (c)!(b): Let U ¼ X n ðd-clðAÞ n AÞ. Then U is dg-open and hence A ¼ U \ d-clðAÞ. (c)!(d): Let F ¼ d-clðAÞ n A. Then X n F ¼ A [ ðX n d-clðAÞÞ and X n F is dg-open and hence A [ ðX n d-clðAÞÞ is dgopen. (d)!(c): Let U ¼ A [ ðX n d-clðAÞÞ. Then X n U is dg-closed and hence X n U ¼ d-clðAÞ n A. h Remark 3.7. However, it is not true that A 2 LdGC ðX ; sÞ if and only if d-intðA [ ðX n d-clðAÞÞÞ A. In fact, let A ¼ fa; bg be a subset of the topological space ðX ; sÞ given in Example 3.4. Then d-intðA [ ðX n d-clðAÞÞÞ ¼ d-intðfa; bgÞ ¼ fag 6 A and A 2 LdGC ðX ; sÞ. Definition 3.8. A topological space ðX ; sÞ is called dg-submaximal (resp. g-submaximal [1]) if every d-dense (resp. dense) subset is dg-open (resp. g-open). Every dg-submaximal is g-submaximal but the converse is not true as seen the following example.
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Example 3.9. Let ðX ; sÞ be the topological space given in Example 3.4. Then ðX ; sÞ g-submaximal but not dg-submaximal, since fa; cg is d-dense (even dense) and g-open but is not dg-open. Proposition 3.10. A topological space ðX ; sÞ is dg-submaximal if and only if P ðX Þ ¼ LdGC ðX ; sÞ. Proof. Let A 2 P ðX Þ and let U ¼ A [ ðX n d-clðAÞÞ. Then X ¼ d-clðU Þ, i.e. U is a d-dense subset of ðX ; sÞ. By assumption, U is dg-open. By Theorem 3.6, U 2 LdGC ðX ; sÞ and hence P ðX Þ ¼ LdGC ðX ; sÞ. Conversely, let A be a d-dense subset of ðX ; sÞ. Then, by assumptions and Theorem 3.6(d), A [ ðX n d-clðAÞÞ ¼ A and A 2 LdGC ðX ; sÞ and hence A is dg-open. This implies ðX ; sÞ is dg-submaximal. h Proposition 3.11. For a subset A of ðX ; sÞ, if A 2 LdGC ðX ; sÞ then there exists a d-open set G such that A ¼ G \ d clðAÞ. Proof. Let A 2 LdGC ðX ; sÞ. Then, there exist a d-open set G and a dg-closed set F such that A ¼ G \ F . Since A G and A d-clðAÞ, A G \ d-clðAÞ and hence A ¼ G \ d-clðAÞ. h The following results are basic properties of ldgc-sets. Proposition 3.12. For subsets A and B of ðX ; sÞ the following are true: (a) If A 2 LdGC ðX ; sÞ and B 2 LdGC ðX ; sÞ then A \ B 2 LdGC ðX ; sÞ. (b) If A 2 LdGC ðX ; sÞ and B is d-closed ðor d-openÞ, then A \ B 2 LdGC ðX ; sÞ. (c) If A 2 LdGCðX ; sÞ and B is dg-open ðor d-closedÞ, then A \ B 2 LdGCðX ; sÞ. Proof. (a) By Theorem 3.6(b), there exist dg-open sets G and U such that A ¼ G \ d-clðAÞ and B ¼ U \ d-clðBÞ. Then, A \ B 2 LdGC ðX ; sÞ since G \ U is dg-open by Lemma 2.4(a) and hence d-clðAÞ \ d-clðBÞ is d-closed.The proofs of (b) and (c) are similar to that of (a). h Proposition 3.13. Let A and B be subsets of ðX ; sÞ, B A and A be open and dg-closed in X . If B is dg-closed in A then B is dg-closed in X . Proof. Let G be an open set in X such that B G. Then B A \ G since A \ G is an open set in A. By assumption, d-clA ðBÞ A \ G. It follows then A \ d-clðBÞ A \ G and A G [ ðX n d-clðBÞÞ. Since A is dg-closed in X , d-clðBÞ d-clðAÞ G [ ðX n d-clðBÞÞ and hence d-clðBÞ G. h Theorem 3.14. Let Z be open subset of ðX ; sÞ and let A Z. (a) If Z is dg-open in ðX ; sÞ and A 2 LdGC ðZ; sjZÞ, then A 2 LdGC ðX ; sÞ. (b) If Z is dg-closed in ðX ; sÞ and A 2 LdGC ðZ; sjZÞ, then A 2 LdGC ðX ; sÞ. (c) If Z is dg-closed and dg-open in ðX ; sÞ and A 2 LdGCðZ; sjZÞ, then A 2 LdGCðX ; sÞ.
Proof (a) By Theorem 3.6, there exists a dg-open set G of ðZ; sjZÞ such that A ¼ G \ d-clZ ðAÞ, where d-clZ ðAÞ ¼ Z \ d-clðAÞ. By Lemma 2.4(a) and Definition 3.1, A ¼ ðZ \ GÞ \ d-clðAÞ 2 LdGC ðX ; sÞ. (b) Let A 2 LdGC ðZ; sjZÞ. Then A ¼ G \ F for some d-open set G and dg-closed set F of ðZ; sjZÞ. By Proposition 3.13 F is dg-closed in ðX ; sÞ. Since G ¼ B \ Z for some d-open set B of ðX ; sÞ, A ¼ ðZ \ BÞ \ F ¼ F \ B. Thus A 2 LdGC ðX ; sÞ. (c) Let A 2 LdGCðZ; sjZÞ. Then A ¼ G \ F for some dg-open set G and dg-closed set F of ðZ; sjZÞ. It is proved that A 2 LdGCðX ; sÞ. h The following example shows that one of the assumptions of Theorem 3.14(a), i.e. Z is dg-open, cannot be removed. Example 3.15. Let X ¼ fa; b; cg and s ¼ fX ; /; fag; fa; bgg. Let V denote the collection of all dg-open sets of ðX ; sÞ. Then we have V ¼ fX; /; fag; fbg; fa; bgg. Putting Z ¼ A ¼ fa; cg. It is shown that Z is not dg-open and A 2 LdGC ðZ; sjZÞ. However, A 62 LdGC ðX ; sÞ since LdGC ðX ; sÞ ¼ P ðX Þ n fa; cg.
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Proposition 3.16. Suppose that the collection of all dg-open sets of ðX ; sÞ is closed under finite unions. Let A 2 LdGC ðX ; sÞ and B 2 LdGC ðX ; sÞ. If A and B are d-separated, i.e. A \ d clðBÞ ¼ / and B \ d clðAÞ ¼ /, then A [ B 2 LdGC ðX ; sÞ. Proof. By Theorem 3.6, there exist dg-open sets G and H of ðX ; sÞ such that A ¼ G \ d-clðAÞ and B ¼ H \ d-clðBÞ. Putting U ¼ G \ ðX n d-clðBÞÞ and V ¼ H \ ðX n d-clðAÞÞ. Then A ¼ U \ d-clðAÞ, B ¼ V \ d-clðBÞ, U \ d-clðBÞ ¼ /, V \ d-clðAÞ ¼ / and U and V are dg-open sets of ðX ; sÞ. Since A [ B ¼ ðU [ V Þ \ ðd-clðA [ BÞÞ and U [ V is dg-open by assumption, we have A [ B 2 LdGC ðX ; sÞ. h The following example shows that one of assumption of Proposition 3.16, i.e. A and B are d-separated, cannot be removed. Example 3.17. Let X ¼ fa; b; c; dg and s ¼ fX ; /; fag; fbg; fa; bg; fa; cg; fa; b; cgg. Then fag 2 LdGC ðX ; sÞ and fdg 2 LdGC ðX ; sÞ. However, fag and fdg are not d-separated and fa; dg 62 LdGC ðX ; sÞ. Proposition 3.18. Let fZi : i 2 Cg be a finite dg-closed cover of ðX ; sÞ, i.e. X ¼ [fZi : i 2 Cg, and let A be a subset of ðX ; sÞ. If A \ Zi 2 LdGC ðZi ; sjZi Þ for each i 2 C, then A 2 LdGC ðX ; sÞ. Proof. For each i 2 C there exist a d-open set Ui of ðX ; sÞ and a dg-closed set Fi of ðZi ; sjZi Þ such that A \ Zi ¼ Ui \ ðZi \ Fi Þ. Then, A ¼ [fA \ Zi : i 2 Cg ¼ ½[fUi : i 2 Cg \ ½[fZi \ Fi : i 2 Cg, and hence A 2 LdGC ðX ; sÞ by Lemma 2.4(a). h
4. LdGC-continuity and LdGC-irresoluteness In [15], Park defined three distinct notions of LdC-continuity, i.e. LdC-irresoluteness, LdC-continuity and sub-LdCcontinuity. In this section we use ldgc-sets, ldgc -sets and ldgc -sets to generalize LdC-irresolute functions, LdCcontinuous functions and sub-LdC-continuous functions and study properties of such functions. Definition 4.1. A function f : ðX ; sÞ ! ðY ; rÞ is called: (a) LdGC-continuous if f 1 ðV Þ 2 LdGCðX ; sÞ for each V 2 r; (b) LdGC -continuous if f 1 ðV Þ 2 LdGC ðX ; sÞ for each V 2 r; (c) LdGC -continuous if f 1 ðV Þ 2 LdGC ðX ; sÞ for each V 2 r. Definition 4.2. A function f : ðX ; sÞ ! ðY ; rÞ is called: (a) LdGC-irresolute if f 1 ðV Þ 2 LdGCðX ; sÞ for each V 2 LdGCðY ; rÞ; (b) LdGC -irresolute if f 1 ðV Þ 2 LdGC ðX ; sÞ for each V 2 LdGC ðY ; rÞ; (c) LdGC -irresolute if f 1 ðV Þ 2 LdGC ðX ; sÞ for each V 2 LdGC ðY ; rÞ. Remark 4.3. For a function f : ðX ; sÞ ! ðY ; rÞ, the following diagram holds which is an enlargement of Balachandran et al. [2] and Park [15]. LdC-continuous LdGC -continuous # GLC -continuous
. ! !
& LdGC-continuous # GLC-continuous
LdGC -continuous # GLC -continuous
In the remark above, the relationships cannot be reversible as the following examples illustrate. Example 4.4. Let X ¼ Y ¼ fa; b; cg and ðX ; sÞ be the topological space given in Example 3.3. and let a function f : ðX ; sÞ ! ðY ; rÞ be the identity. (a) If r ¼ fX ; /; fcgg is a topology on Y , then f is LdGC -continuous but it is not LdC-continuous, since f 1 ðfcgÞ 2 LdGC ðX ; sÞ but f 1 ðfcgÞ 62 LdCðX ; sÞ.
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(b) If r ¼ fX ; /; fa; bgg is a topology on Y , then f is LdGC -continuous but it is not LdC-continuous, f 1 ðfa; bgÞ 2 LdGC ðX ; sÞ but f 1 ðfa; bgÞ 62 LdCðX ; sÞ. (c) If r ¼ fX ; /; fcgg is a topology on Y , then f is GLC -continuous but it is not LdGC -continuous, f 1 ðfcgÞ 2 GLC ðX ; sÞ but f 1 ðfcgÞ 62 LdGC ðX ; sÞ. (d) If r ¼ fX ; /; fcgg is a topology on of Y , then f is LdGC-continuous but it is not LdGC -continuous, f 1 ðfcgÞ 2 LdGCðX ; sÞ but f 1 ðfcgÞ 62 LdGC ðX ; sÞ. (e) If r ¼ fX ; /; fa; bgg is a topology on Y , then f is LdGC-continuous but it is not LdGC -continuous, f 1 ðfa; bgÞ 2 LdGCðX ; sÞ but f 1 ðfa; bgÞ 62 LdGC ðX ; sÞ.
since since since since
Remark 4.5. For a function f : ðX ; sÞ ! ðY ; rÞ, the following statements hold: (a) If a space ðX ; sÞ is semi-regular, then LdGC -continuous, LdGC -continuous and GLC-continuous coincide. (b) If a space ðX ; sÞ is T3=4 , then LdGC-continuous and LdC-continuous coincide. Proposition 4.6. A topological space ðX ; sÞ is dg-submaximal if and only if every function having ðX ; sÞ as its domain is LdGC -continuous. Proof. Suppose that f : ðX ; sÞ ! ðY ; rÞ is a function. By Proposition 3.10 we have that f 1 ðV Þ 2 LdGC ðX ; sÞ ¼ P ðX Þ for each open set V of ðY ; rÞ. Therefore, f is LdGC -continuous. Conversely, let Y ¼ f0; 1g be the Sierpinski space with topology r ¼ fY ; /; f0gg. Let V be a subset of ðX ; sÞ and f : ðX ; sÞ ! ðY ; rÞ a function defined by f ðxÞ ¼ 0 for every x 2 V and f ðxÞ ¼ 1 for every x 62 V . By assumption, f is LdGC -continuous and hence f 1 ðf0gÞ ¼ V 2 LdGC ðX ; sÞ. Therefore we have P ðX Þ ¼ LdGC ðX ; sÞ and so ðX ; sÞ is dg-submaximal by Proposition 3.10. h Proposition 4.7. If f : ðX ; sÞ ! ðY ; rÞ is LdGC -continuous and a subset B is d-closed in ðX ; sÞ, then the restriction of f to B, say f jB : ðB; sjBÞ ! ðY ; rÞ is LdGC -continuous. Proof. Let V be an open set of ðY ; rÞ. Then, f 1 ðV Þ ¼ G \ F for some d-open set G and dg-closed set F of ðX ; sÞ. By Proposition 3.13, ðf jBÞ1 ðV Þ ¼ ðG \ BÞ \ ðF \ BÞ 2 LdGC ðB; sjBÞ. This implies that f jB is LdGC -continuous. h We recall the definition of the combination of two functions: let X ¼ A [ B and f : A ! Y and h : B ! Y be two functions. We say that f and h are compatible if f jA \ B ¼ hjA \ B. Then, we can define a function f rh : X ! Y as follows: ðf rhÞðxÞ ¼ f ðxÞ for every x 2 A and ðf rhÞðxÞ ¼ hðxÞ for every x 2 B. The function f rh : X ! Y is called the combination of f and h. Theorem 4.8. Let X ¼ A [ B, where A and B are dg-closed sets of ðX ; sÞ, and f : ðA; sjAÞ ! ðY ; rÞ and h : ðB; sjBÞ ! ðY ; rÞ be compatible functions. (a) If f and h are LdGC -continuous, then f rh : ðX ; sÞ ! ðY ; rÞ is LdGC -continuous. (b) If f and h are LdGC -irresolute, then f rh : ðX ; sÞ ! ðY ; rÞ is LdGC -irresolute. Proof (a) Let V 2 r. Then, ðf rhÞ1 ðV Þ \ A ¼ f 1 ðV Þ and ðf rhÞ1 ðV Þ \ B ¼ h1 ðV Þ. By assumptions, ðf rhÞ1 ðV Þ \ A 2 LdGC ðA; sjAÞ and ðf rhÞ1 ðV Þ \ B 2 LdGC ðB; sjBÞ. Therefore, by Proposition 3.18, ðf rhÞ1 ðV Þ 2 LdGC ðX ; sÞ and hence f rh is LdGC -continuous. (b) The proof is similar to (a). h Next we offer the following compositions theorem: Theorem 4.9. Let f : ðX ; sÞ ! ðY ; rÞ and g : ðY ; rÞ ! ðZ; gÞ be the functions. (a) The composition f g of two LdGC-irresolute (resp. LdGC -irresolute, LdGC -irresolute) functions are LdGC-irresolute (resp. LdGC -irresolute, LdGC -irresolute). (b) If f is LdGC-irresolute(resp. LdGC -irresolute, LdGC -irresolute) and g is LdGC-continuous (resp. LdGC -continuous, LdGC -continuous), then the composition f g is LdGC-continuous (resp. LdGC -continuous, LdGC -continuous). Proof. It is immediate from Definition 4.1 and 4.2. h
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In the end of this section we introduce the notion of sub-LdC-continuity which is properly placed between sub-LdCcontinuity [15] and sub-GLC -continuity [1]. Definition 4.10. A function f : ðX ; sÞ ! ðY ; rÞ is called: (a) sub-GLC -continuous [1] if there is a basis B for ðY ; rÞ such that f 1 ðU Þ 2 GLC ðX ; sÞ for each U 2 B; (b) sub-LdC-continuous [15] if there is a basis B for ðY ; rÞ such that f 1 ðU Þ 2 LdCðX ; sÞ for each U 2 B; (c) sub-LdGC -continuous if there is a basis B for ðY ; rÞ such that f 1 ðU Þ 2 LdGC ðX ; sÞ for each U 2 B. Proposition 4.11. A function f : ðX ; sÞ ! ðY ; rÞ is sub-LdGC -continuous if and only if there is a subbasis C of ðY ; rÞ such that f 1 ðU Þ 2 LdGC ðX ; sÞ for each U 2 C. Proof. By assumption, there exists a basis B for ðY ; rÞ such that f 1 ðU Þ 2 LdGC ðX ; sÞ for each U 2 B. Since B is also a subbasis for ðY ; rÞ, the proof is obvious. Conversely, for a subbasis C, let Cd ¼ fA Y : A is an intersection of finitely many sets belonging to Cg. Then, Cd is a basis for ðY ; rÞ. For U 2 Cd , U ¼ \fFi : Fi 2 C; i 2 Kg where K is a finite set, By assumption and Proposition 3.12, we have f 1 ðU Þ ¼ \ff 1 ðFi Þ : i 2 Kg 2 LdGC ðX ; sÞ. h Remark 4.12. For a function f : ðX ; sÞ ! ðY ; rÞ, the following implications hold: LdC-continuous ! # sub-LdC-continuous !
LdGC -continuous # sub-LdGC -continuous
! !
GLC -continuous # sub-GLC -continuous
In the remark above, the relationships cannot be reversible as the following examples illustrate. Example 4.13. Let X ¼ Y ¼ fa; b; cg and s ¼ fX ; /; fag; fcg; fa; cg; fb; cgg and r be the topology induced by a base B of Y . Let f : ðX ; sÞ ! ðY ; rÞ be the identity function. (a) If B ¼ fY ; fag; fcgg, then f is sub-LdGC -continuous, but it is not dGLC -continuous, since f 1 ðfa; cgÞ 62 LdGC ðX ; sÞ for fa; cg 2 r. (b) If B ¼ fY ; fcgg, then f is sub-LdGC -continuous, but it is not sub-LdC-continuous, since f 1 ðfcgÞ ¼ fcg 2 LdGC ðX ; sÞ but f 1 ðfcgÞ 62 LdCðX ; sÞ. (c) If B ¼ fY ; fag; fcg; fa; cgg, then f is sub-GLC -continuous, but it is not sub-LdGC -continuous, since f 1 ðfa; cgÞ 2 GLC ðX ; sÞ but f 1 ðfa; cgÞ 62 LdGC ðX ; sÞ. The following result is an immediate consequence of Remark 3.5. Remark 4.14. The following statements hold for any f : ðX ; sÞ ! ðY ; rÞ: (a) If a space ðX ; sÞ is semi-regular, then sub-LdGC -continuous and sub-GLC -continuous coincide. (b) If a space ðX ; sÞ is T3=4 , then sub-LdGC-continuous and sub-LdC -continuous coincide. In conclusion, generalizations of closed sets in point-set topology suggest some new separation axioms which have been found to be very useful in the study of certain objects of digital topology. Thus we may stress once more the importance of locally d-generalized closed sets as a branch of them and the possible application in computer graphics [8,9] and quantum physics [4]. Acknowledgement This work is supported by grant no. R05-2002-000-01045-0 from the Korea Science & Engineering Foundation. References [1] Balachandran K, Sundaram P, Maki H. Generalized locally closed sets and GLC-continuous functions. Indian J Pure Appl Math 1996;27:235–44.
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[2] Bourbaki N. General topology, Part 1. Reading. MA: Addison Wesley; 1966. [3] Dontchev J, Ganster M. On d-generalized closed sets and T3 =4-spaces. Mem Fac Sci Kochi Univ Ser (Math) 1996;17:15–31. [4] Elnaschie MS. On the uncertainty of Cantorian geometry and the two-slit experiment. Chaos, Solitons & Fractals 1998;9(3):517– 29. [5] Ganster M, Reilly IL. Locally closed sets and LC-continuous functions. Int J Math Math Sci 1989;12:417–24. [6] Ganster M, Reilly IL, Vamanamurthy MK. Remarks on locally closed sets. Math Pannonica 1992;3(2):107–13. [7] Khalimsky ED. Applications of connected ordered topological spaces in topology. Conference of Math Departments of Povolsia, 1970. [8] Khalimsky ED, Kopperman R, Meyer PR. Computer graphics and connected topologies on finite ordered sets. Topol Appl 1990;36:1–17. [9] Kong TY, Kopperman R, Meyer PR. A topological approach to digital topology. Am Math Monthly 1991;98:901–17. [10] Kovalevsky V, Kopperman R. Some topology-based image processing algorithms. Ann NY Acad Sci 1994;728:174–82. [11] Kuratowski C, Sierpinski W. Sur les differences de deux ensemble fermes. Tohuku Math J 1921;20:22–5. [12] Mrsevic M, Reilly IL, Vamanamurthy MK. On semi-regularization properties. J Austral Math Soc (Series A) 1985;38:40–54. [13] Levine N. Generalized closed sets in topology. Rend Cir Mat Palermo 1970;19:55–60. [14] Nasef AA. On b-locally closed sets and related topics. Chaos, Solitons & Fractals 2001;12:1910–5. [15] Park JH. Locally d-closed sets and LdC-continuous functions. Indian J Pure Appl Math, submitted. [16] Park JH, Park JK. On semi generalized locally closed sets and SGLC-continuous functions. Indian J Pure Appl Math 2000; 31(9):1103–12. [17] Velicko NV. H -closed topological spaces. Am Math Soc Trans 1968;78:103–18.
On locally d-generalized closed sets and LdGC-continuous functions Jin Keun Park a, Jin Han Park a
a,*
, Bu Young Lee
b
Division of Mathematical Sciences, Pukyong National University, 599-1, Daeyeon 3-Dong, Nam-Gu, Pusan 608-737, South Korea b Departments of Mathematics, Dong-A University, 840, Hadan 2-Dong, Saha-Gu, Pusan 604-714, South Korea Accepted 28 April 2003
Abstract In this paper we introduce ldgc-sets, ldgc -sets and ldgc -sets and different notions of generalizations of continuous functions in a topological space and study some of their relationship and their properties. Possible application to computer graphics and quantum physics are touched upon. Ó 2003 Elsevier Ltd. All rights reserved.
1. Introduction In 1968, Velicko [17] introduced d-open sets in order to investigate the characterization of H-closed spaces and showed that sd , i.e. the collection of all d-open sets, is the topology on X such that sd s and, in particular, sd equal to the semi-regularization topology ss . The approach of the concept of semi-regularization from a different perspective via the concept of generalized closedness was done by Dontchev and Ganster [3]. They consider sets whose closure in the semi-regularization are contained in every superset which is open in the original topology. They called these sets dgeneralized closed (briefly, dg-closed) and studied their basic properties. The notion of a locally closed set in a topological space was implicitly introduced by Kuratowski and Sierpienski [11]. According to Bourbaki [2] a subset of a topological space X is locally closed in X if it is the intersection of an open set and a closed set in X . In 1989, Ganster and Reilly [6] continued the study of locally closed sets and also introduced the concept of LC-continuous functions to find a decomposition of continuous functions. Thereafter, intensive research on the field of locally closed sets and LCcontinuous functions has been done as theory developed by Balachandran et al. [1], Ganster et al. [6], Nasef [14] and Park and Park [16]. Several topological spaces that fail to be T1 are very often of significant importance in digital topology, i.e. in the study of the geometric and topological properties of digital images [9,10]. Such is the case with the major building block of the digital n-space, the digital line or the so called Khalimsky line [7–10]. This is the set of the integers, Z, equipped with the topology K, generated by GK ¼ ff2n 1; 2n; 2n þ 1g : n 2 Zg. The Khalimsky line, a space which is widely used in the applications of point-set topology in computer graphics, is T3=4 but not T1 [3]. The purpose of this paper is to introduce three new classes of sets called ldgc-sets, ldgc -sets and ldgc -sets, which are contained in the class of glc-sets and contain the class of ldc-sets by using the notions of d-open and d-closed sets due to Velicko [17] and dg-open and dg-closed sets due to Dontchev and Ganster [3]. Also, we introduce some different classes of continuity and irresoluteness and study some of their properties.
*
Corresponding author. E-mail address: [email protected] (J.H. Park).
0960-0779/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0960-0779(03)00194-2
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2. Preliminaries All topological spaces considered in this paper lack any separation axioms unless explicitly stated. The topological space is denoted by ðX ; sÞ will be replaced by X if there is no chance for confusion. For a subset A X , the closure and the interior of A in X are denoted by clðAÞ and intðAÞ, respectively. The d-interior [17] of a subset A of X is the union of all regular open sets of X contained in A and is denoted by d-intðAÞ. A subset A is called d-open [17] if A ¼ d-intðAÞ, i.e. a set is d-open, if it is the union of regular open sets. The complement of a d-open set is called d-closed. Alternatively, a set A ðX ; sÞ is called d-closed [17] if A ¼ d-clðAÞ, where d-clðAÞ ¼ fx 2 X : intðClðU ÞÞ \ A 6¼ /; for every open set U containing xg. The family of all d-open sets form a topology on X and is denoted sd . Recall the following definitions which will be used in sequel. Definition 2.1. A subset A of a space X is called: (a) g-closed [13] if clðAÞ G whenever A G and G is open in X ; (b) dg-closed [3] if d-clðAÞ G whenever A G and G is open in X ; (c) g-open (resp. dg-open) if the complement of A is g-closed (resp. dg-closed).
Definition 2.2. A subset A of ðX ; sÞ is called: (a) (b) (c) (d) (e)
locally closed set [5] if A ¼ G \ F where G 2 s and F is closed in ðX ; sÞ; locally d-closed set [15] (briefly, ldc-set) if A ¼ G \ F where G is d-open and F is d-closed in ðX ; sÞ; generalized locally closed set [1] (briefly, glc-set) if A ¼ G \ F where G is g-open in ðX ; sÞ and F is g-closed in ðX ; sÞ; glc -set [1] if there exist a g-open set G and a closed set F of ðX ; sÞ such that A ¼ G \ F ; glc -set [1] if there exist an open set G and a g-closed set F of ðX ; sÞ such that A ¼ G \ F .
The collection of all locally d-closed sets (resp. glc-sets, glc -sets, glc -sets) of ðX ; sÞ will be denoted by LdCðX ; sÞ (resp. GLCðX ; sÞ, GLC ðX ; sÞ, GLC ðX ; sÞ.) Definition 2.3. A function f : ðX ; sÞ ! ðY ; rÞ is called: (a) (b) (c) (d)
LdC-continuous [14] if f 1 ðV Þ 2 LdCðX ; sÞ for each V 2 r; GLC -continuous [1] if f 1 ðV Þ 2 GLC ðX ; sÞ for each V 2 r; GLC -continuous [1] if f 1 ðV Þ 2 GLC ðX ; sÞ for each V 2 r; GLC-continuous [1] if f 1 ðV Þ 2 GLCðX ; sÞ for each V 2 r.
Lemma 2.4 [3] (a) Finite union of dg-closed sets is a dg-closed set. (b) The intersection of a dg-closed set and a d-closed set is dg-closed.
3. Locally d-generalized closed sets We begin with the following definitions as a generalization of ldc-sets. Definition 3.1. A subset A of ðX ; sÞ is called: (a) locally d-generalized closed set (briefly, ldgc-set) if there exist a dg-open set G and a dg-closed set F of ðX ; sÞ such that A ¼ G \ F ; (b) ldgc -set if there exist a dg-open set G and a d-closed set F of ðX ; sÞ such that A ¼ G \ F ; (c) ldgc -set if there exist a d-open set G and a dg-closed set F of ðX ; sÞ such that A ¼ G \ F . The collection of all ldgc-sets (resp. ldgc -sets, ldgc -sets) of ðX ; sÞ will be denoted by LdGCðX ; sÞ (resp. LdGC ðX ; sÞ, LdGC ðX ; sÞ).
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Remark 3.2. From the definitions above and Definition 2.2, we have the following implications: ldgc -set %
! &
ldc-set
glc -set ldgc-set
& ldgc -set
% !
& ! %
glc-set
glc -set
In the remark above, the relationships cannot be reversible as the following examples illustrate. Example 3.3. Let X ¼ fa; b; cg and s ¼ fX ; /; fag; fcg; fa; cg; fb; cgg. (a) fcg 2 LdGC ðX ; sÞ but fcg 62 LdCðX ; sÞ. (b) fa; bg 2 LdGC ðX ; sÞ but fa; bg 62 LdCðX ; sÞ. (c) fa; bg 2 LdGCðX ; sÞ but fa; bg 62 LdGC ðX ; sÞ. (d) fcg 2 LdGCðX ; sÞ but fcg 62 LdGC ðX ; sÞ. (e) fa; cg 2 GLC ðX ; sÞ but fa; cg 62 LdGC ðX ; sÞ. (f) fcg 2 GLC ðX ; sÞ but fcg 62 LdGC ðX ; sÞ. Example 3.4. Let X ¼ fa; b; c; dg and s ¼ fX ; /; fag; fbg; fa; bg; fa; b; cg; fa; b; dgg. Then fcg 2 GLCðX ; sÞ but fcg 62 LdGCðX ; sÞ. Recall that a topological space ðX ; sÞ is called semi-regular [12] if s ¼ ss and a topological space ðX ; sÞ is called T3=4 [3] if every dg-closed set is d-closed in ðX ; sÞ. Remark 3.5 (a) If a space ðX ; sÞ is semi-regular, then LdGC ðX ; sÞ ¼ LdGC ðX ; sÞ ¼ GLCðX ; sÞ: (b) If a space ðX ; sÞ is T3=4 , then LdGCðX ; sÞ ¼ LdCðX ; sÞ: Now, we obtain a characterization for ldgc -sets as follows: Theorem 3.6. For a subset A of ðX ; sÞ, the following are equivalent: (a) (b) (c) (d)
A 2 LdGC ðX ; sÞ. A ¼ G \ d-clðAÞ for some dg-open set G. d-clðAÞ n A is dg-closed. A [ ðX n d-clðAÞÞ is dg-open.
Proof (a)!(b): Let A 2 LdGC ðX ; sÞ. Then A ¼ G \ F , where G is dg-open and F is d-closed. A F implies d-clðAÞ F and so A ¼ G \ F G \ d-clðAÞ. And A G and A d-clðAÞ implies A G \ d-clðAÞ. Hence A ¼ G \ d-clðAÞ. (b)!(a): Since G is dg-open and d-clðAÞ is d-closed, G \ d-clðAÞ 2 LdGC ðX ; sÞ by Definition 3.1(b). (b)!(c): d-clðAÞ n A ¼ d-clðAÞ \ ðX n GÞ, which is dg-closed by Lemma 2.4(b). (c)!(b): Let U ¼ X n ðd-clðAÞ n AÞ. Then U is dg-open and hence A ¼ U \ d-clðAÞ. (c)!(d): Let F ¼ d-clðAÞ n A. Then X n F ¼ A [ ðX n d-clðAÞÞ and X n F is dg-open and hence A [ ðX n d-clðAÞÞ is dgopen. (d)!(c): Let U ¼ A [ ðX n d-clðAÞÞ. Then X n U is dg-closed and hence X n U ¼ d-clðAÞ n A. h Remark 3.7. However, it is not true that A 2 LdGC ðX ; sÞ if and only if d-intðA [ ðX n d-clðAÞÞÞ A. In fact, let A ¼ fa; bg be a subset of the topological space ðX ; sÞ given in Example 3.4. Then d-intðA [ ðX n d-clðAÞÞÞ ¼ d-intðfa; bgÞ ¼ fag 6 A and A 2 LdGC ðX ; sÞ. Definition 3.8. A topological space ðX ; sÞ is called dg-submaximal (resp. g-submaximal [1]) if every d-dense (resp. dense) subset is dg-open (resp. g-open). Every dg-submaximal is g-submaximal but the converse is not true as seen the following example.
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Example 3.9. Let ðX ; sÞ be the topological space given in Example 3.4. Then ðX ; sÞ g-submaximal but not dg-submaximal, since fa; cg is d-dense (even dense) and g-open but is not dg-open. Proposition 3.10. A topological space ðX ; sÞ is dg-submaximal if and only if P ðX Þ ¼ LdGC ðX ; sÞ. Proof. Let A 2 P ðX Þ and let U ¼ A [ ðX n d-clðAÞÞ. Then X ¼ d-clðU Þ, i.e. U is a d-dense subset of ðX ; sÞ. By assumption, U is dg-open. By Theorem 3.6, U 2 LdGC ðX ; sÞ and hence P ðX Þ ¼ LdGC ðX ; sÞ. Conversely, let A be a d-dense subset of ðX ; sÞ. Then, by assumptions and Theorem 3.6(d), A [ ðX n d-clðAÞÞ ¼ A and A 2 LdGC ðX ; sÞ and hence A is dg-open. This implies ðX ; sÞ is dg-submaximal. h Proposition 3.11. For a subset A of ðX ; sÞ, if A 2 LdGC ðX ; sÞ then there exists a d-open set G such that A ¼ G \ d clðAÞ. Proof. Let A 2 LdGC ðX ; sÞ. Then, there exist a d-open set G and a dg-closed set F such that A ¼ G \ F . Since A G and A d-clðAÞ, A G \ d-clðAÞ and hence A ¼ G \ d-clðAÞ. h The following results are basic properties of ldgc-sets. Proposition 3.12. For subsets A and B of ðX ; sÞ the following are true: (a) If A 2 LdGC ðX ; sÞ and B 2 LdGC ðX ; sÞ then A \ B 2 LdGC ðX ; sÞ. (b) If A 2 LdGC ðX ; sÞ and B is d-closed ðor d-openÞ, then A \ B 2 LdGC ðX ; sÞ. (c) If A 2 LdGCðX ; sÞ and B is dg-open ðor d-closedÞ, then A \ B 2 LdGCðX ; sÞ. Proof. (a) By Theorem 3.6(b), there exist dg-open sets G and U such that A ¼ G \ d-clðAÞ and B ¼ U \ d-clðBÞ. Then, A \ B 2 LdGC ðX ; sÞ since G \ U is dg-open by Lemma 2.4(a) and hence d-clðAÞ \ d-clðBÞ is d-closed.The proofs of (b) and (c) are similar to that of (a). h Proposition 3.13. Let A and B be subsets of ðX ; sÞ, B A and A be open and dg-closed in X . If B is dg-closed in A then B is dg-closed in X . Proof. Let G be an open set in X such that B G. Then B A \ G since A \ G is an open set in A. By assumption, d-clA ðBÞ A \ G. It follows then A \ d-clðBÞ A \ G and A G [ ðX n d-clðBÞÞ. Since A is dg-closed in X , d-clðBÞ d-clðAÞ G [ ðX n d-clðBÞÞ and hence d-clðBÞ G. h Theorem 3.14. Let Z be open subset of ðX ; sÞ and let A Z. (a) If Z is dg-open in ðX ; sÞ and A 2 LdGC ðZ; sjZÞ, then A 2 LdGC ðX ; sÞ. (b) If Z is dg-closed in ðX ; sÞ and A 2 LdGC ðZ; sjZÞ, then A 2 LdGC ðX ; sÞ. (c) If Z is dg-closed and dg-open in ðX ; sÞ and A 2 LdGCðZ; sjZÞ, then A 2 LdGCðX ; sÞ.
Proof (a) By Theorem 3.6, there exists a dg-open set G of ðZ; sjZÞ such that A ¼ G \ d-clZ ðAÞ, where d-clZ ðAÞ ¼ Z \ d-clðAÞ. By Lemma 2.4(a) and Definition 3.1, A ¼ ðZ \ GÞ \ d-clðAÞ 2 LdGC ðX ; sÞ. (b) Let A 2 LdGC ðZ; sjZÞ. Then A ¼ G \ F for some d-open set G and dg-closed set F of ðZ; sjZÞ. By Proposition 3.13 F is dg-closed in ðX ; sÞ. Since G ¼ B \ Z for some d-open set B of ðX ; sÞ, A ¼ ðZ \ BÞ \ F ¼ F \ B. Thus A 2 LdGC ðX ; sÞ. (c) Let A 2 LdGCðZ; sjZÞ. Then A ¼ G \ F for some dg-open set G and dg-closed set F of ðZ; sjZÞ. It is proved that A 2 LdGCðX ; sÞ. h The following example shows that one of the assumptions of Theorem 3.14(a), i.e. Z is dg-open, cannot be removed. Example 3.15. Let X ¼ fa; b; cg and s ¼ fX ; /; fag; fa; bgg. Let V denote the collection of all dg-open sets of ðX ; sÞ. Then we have V ¼ fX; /; fag; fbg; fa; bgg. Putting Z ¼ A ¼ fa; cg. It is shown that Z is not dg-open and A 2 LdGC ðZ; sjZÞ. However, A 62 LdGC ðX ; sÞ since LdGC ðX ; sÞ ¼ P ðX Þ n fa; cg.
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Proposition 3.16. Suppose that the collection of all dg-open sets of ðX ; sÞ is closed under finite unions. Let A 2 LdGC ðX ; sÞ and B 2 LdGC ðX ; sÞ. If A and B are d-separated, i.e. A \ d clðBÞ ¼ / and B \ d clðAÞ ¼ /, then A [ B 2 LdGC ðX ; sÞ. Proof. By Theorem 3.6, there exist dg-open sets G and H of ðX ; sÞ such that A ¼ G \ d-clðAÞ and B ¼ H \ d-clðBÞ. Putting U ¼ G \ ðX n d-clðBÞÞ and V ¼ H \ ðX n d-clðAÞÞ. Then A ¼ U \ d-clðAÞ, B ¼ V \ d-clðBÞ, U \ d-clðBÞ ¼ /, V \ d-clðAÞ ¼ / and U and V are dg-open sets of ðX ; sÞ. Since A [ B ¼ ðU [ V Þ \ ðd-clðA [ BÞÞ and U [ V is dg-open by assumption, we have A [ B 2 LdGC ðX ; sÞ. h The following example shows that one of assumption of Proposition 3.16, i.e. A and B are d-separated, cannot be removed. Example 3.17. Let X ¼ fa; b; c; dg and s ¼ fX ; /; fag; fbg; fa; bg; fa; cg; fa; b; cgg. Then fag 2 LdGC ðX ; sÞ and fdg 2 LdGC ðX ; sÞ. However, fag and fdg are not d-separated and fa; dg 62 LdGC ðX ; sÞ. Proposition 3.18. Let fZi : i 2 Cg be a finite dg-closed cover of ðX ; sÞ, i.e. X ¼ [fZi : i 2 Cg, and let A be a subset of ðX ; sÞ. If A \ Zi 2 LdGC ðZi ; sjZi Þ for each i 2 C, then A 2 LdGC ðX ; sÞ. Proof. For each i 2 C there exist a d-open set Ui of ðX ; sÞ and a dg-closed set Fi of ðZi ; sjZi Þ such that A \ Zi ¼ Ui \ ðZi \ Fi Þ. Then, A ¼ [fA \ Zi : i 2 Cg ¼ ½[fUi : i 2 Cg \ ½[fZi \ Fi : i 2 Cg, and hence A 2 LdGC ðX ; sÞ by Lemma 2.4(a). h
4. LdGC-continuity and LdGC-irresoluteness In [15], Park defined three distinct notions of LdC-continuity, i.e. LdC-irresoluteness, LdC-continuity and sub-LdCcontinuity. In this section we use ldgc-sets, ldgc -sets and ldgc -sets to generalize LdC-irresolute functions, LdCcontinuous functions and sub-LdC-continuous functions and study properties of such functions. Definition 4.1. A function f : ðX ; sÞ ! ðY ; rÞ is called: (a) LdGC-continuous if f 1 ðV Þ 2 LdGCðX ; sÞ for each V 2 r; (b) LdGC -continuous if f 1 ðV Þ 2 LdGC ðX ; sÞ for each V 2 r; (c) LdGC -continuous if f 1 ðV Þ 2 LdGC ðX ; sÞ for each V 2 r. Definition 4.2. A function f : ðX ; sÞ ! ðY ; rÞ is called: (a) LdGC-irresolute if f 1 ðV Þ 2 LdGCðX ; sÞ for each V 2 LdGCðY ; rÞ; (b) LdGC -irresolute if f 1 ðV Þ 2 LdGC ðX ; sÞ for each V 2 LdGC ðY ; rÞ; (c) LdGC -irresolute if f 1 ðV Þ 2 LdGC ðX ; sÞ for each V 2 LdGC ðY ; rÞ. Remark 4.3. For a function f : ðX ; sÞ ! ðY ; rÞ, the following diagram holds which is an enlargement of Balachandran et al. [2] and Park [15]. LdC-continuous LdGC -continuous # GLC -continuous
. ! !
& LdGC-continuous # GLC-continuous
LdGC -continuous # GLC -continuous
In the remark above, the relationships cannot be reversible as the following examples illustrate. Example 4.4. Let X ¼ Y ¼ fa; b; cg and ðX ; sÞ be the topological space given in Example 3.3. and let a function f : ðX ; sÞ ! ðY ; rÞ be the identity. (a) If r ¼ fX ; /; fcgg is a topology on Y , then f is LdGC -continuous but it is not LdC-continuous, since f 1 ðfcgÞ 2 LdGC ðX ; sÞ but f 1 ðfcgÞ 62 LdCðX ; sÞ.
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(b) If r ¼ fX ; /; fa; bgg is a topology on Y , then f is LdGC -continuous but it is not LdC-continuous, f 1 ðfa; bgÞ 2 LdGC ðX ; sÞ but f 1 ðfa; bgÞ 62 LdCðX ; sÞ. (c) If r ¼ fX ; /; fcgg is a topology on Y , then f is GLC -continuous but it is not LdGC -continuous, f 1 ðfcgÞ 2 GLC ðX ; sÞ but f 1 ðfcgÞ 62 LdGC ðX ; sÞ. (d) If r ¼ fX ; /; fcgg is a topology on of Y , then f is LdGC-continuous but it is not LdGC -continuous, f 1 ðfcgÞ 2 LdGCðX ; sÞ but f 1 ðfcgÞ 62 LdGC ðX ; sÞ. (e) If r ¼ fX ; /; fa; bgg is a topology on Y , then f is LdGC-continuous but it is not LdGC -continuous, f 1 ðfa; bgÞ 2 LdGCðX ; sÞ but f 1 ðfa; bgÞ 62 LdGC ðX ; sÞ.
since since since since
Remark 4.5. For a function f : ðX ; sÞ ! ðY ; rÞ, the following statements hold: (a) If a space ðX ; sÞ is semi-regular, then LdGC -continuous, LdGC -continuous and GLC-continuous coincide. (b) If a space ðX ; sÞ is T3=4 , then LdGC-continuous and LdC-continuous coincide. Proposition 4.6. A topological space ðX ; sÞ is dg-submaximal if and only if every function having ðX ; sÞ as its domain is LdGC -continuous. Proof. Suppose that f : ðX ; sÞ ! ðY ; rÞ is a function. By Proposition 3.10 we have that f 1 ðV Þ 2 LdGC ðX ; sÞ ¼ P ðX Þ for each open set V of ðY ; rÞ. Therefore, f is LdGC -continuous. Conversely, let Y ¼ f0; 1g be the Sierpinski space with topology r ¼ fY ; /; f0gg. Let V be a subset of ðX ; sÞ and f : ðX ; sÞ ! ðY ; rÞ a function defined by f ðxÞ ¼ 0 for every x 2 V and f ðxÞ ¼ 1 for every x 62 V . By assumption, f is LdGC -continuous and hence f 1 ðf0gÞ ¼ V 2 LdGC ðX ; sÞ. Therefore we have P ðX Þ ¼ LdGC ðX ; sÞ and so ðX ; sÞ is dg-submaximal by Proposition 3.10. h Proposition 4.7. If f : ðX ; sÞ ! ðY ; rÞ is LdGC -continuous and a subset B is d-closed in ðX ; sÞ, then the restriction of f to B, say f jB : ðB; sjBÞ ! ðY ; rÞ is LdGC -continuous. Proof. Let V be an open set of ðY ; rÞ. Then, f 1 ðV Þ ¼ G \ F for some d-open set G and dg-closed set F of ðX ; sÞ. By Proposition 3.13, ðf jBÞ1 ðV Þ ¼ ðG \ BÞ \ ðF \ BÞ 2 LdGC ðB; sjBÞ. This implies that f jB is LdGC -continuous. h We recall the definition of the combination of two functions: let X ¼ A [ B and f : A ! Y and h : B ! Y be two functions. We say that f and h are compatible if f jA \ B ¼ hjA \ B. Then, we can define a function f rh : X ! Y as follows: ðf rhÞðxÞ ¼ f ðxÞ for every x 2 A and ðf rhÞðxÞ ¼ hðxÞ for every x 2 B. The function f rh : X ! Y is called the combination of f and h. Theorem 4.8. Let X ¼ A [ B, where A and B are dg-closed sets of ðX ; sÞ, and f : ðA; sjAÞ ! ðY ; rÞ and h : ðB; sjBÞ ! ðY ; rÞ be compatible functions. (a) If f and h are LdGC -continuous, then f rh : ðX ; sÞ ! ðY ; rÞ is LdGC -continuous. (b) If f and h are LdGC -irresolute, then f rh : ðX ; sÞ ! ðY ; rÞ is LdGC -irresolute. Proof (a) Let V 2 r. Then, ðf rhÞ1 ðV Þ \ A ¼ f 1 ðV Þ and ðf rhÞ1 ðV Þ \ B ¼ h1 ðV Þ. By assumptions, ðf rhÞ1 ðV Þ \ A 2 LdGC ðA; sjAÞ and ðf rhÞ1 ðV Þ \ B 2 LdGC ðB; sjBÞ. Therefore, by Proposition 3.18, ðf rhÞ1 ðV Þ 2 LdGC ðX ; sÞ and hence f rh is LdGC -continuous. (b) The proof is similar to (a). h Next we offer the following compositions theorem: Theorem 4.9. Let f : ðX ; sÞ ! ðY ; rÞ and g : ðY ; rÞ ! ðZ; gÞ be the functions. (a) The composition f g of two LdGC-irresolute (resp. LdGC -irresolute, LdGC -irresolute) functions are LdGC-irresolute (resp. LdGC -irresolute, LdGC -irresolute). (b) If f is LdGC-irresolute(resp. LdGC -irresolute, LdGC -irresolute) and g is LdGC-continuous (resp. LdGC -continuous, LdGC -continuous), then the composition f g is LdGC-continuous (resp. LdGC -continuous, LdGC -continuous). Proof. It is immediate from Definition 4.1 and 4.2. h
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In the end of this section we introduce the notion of sub-LdC-continuity which is properly placed between sub-LdCcontinuity [15] and sub-GLC -continuity [1]. Definition 4.10. A function f : ðX ; sÞ ! ðY ; rÞ is called: (a) sub-GLC -continuous [1] if there is a basis B for ðY ; rÞ such that f 1 ðU Þ 2 GLC ðX ; sÞ for each U 2 B; (b) sub-LdC-continuous [15] if there is a basis B for ðY ; rÞ such that f 1 ðU Þ 2 LdCðX ; sÞ for each U 2 B; (c) sub-LdGC -continuous if there is a basis B for ðY ; rÞ such that f 1 ðU Þ 2 LdGC ðX ; sÞ for each U 2 B. Proposition 4.11. A function f : ðX ; sÞ ! ðY ; rÞ is sub-LdGC -continuous if and only if there is a subbasis C of ðY ; rÞ such that f 1 ðU Þ 2 LdGC ðX ; sÞ for each U 2 C. Proof. By assumption, there exists a basis B for ðY ; rÞ such that f 1 ðU Þ 2 LdGC ðX ; sÞ for each U 2 B. Since B is also a subbasis for ðY ; rÞ, the proof is obvious. Conversely, for a subbasis C, let Cd ¼ fA Y : A is an intersection of finitely many sets belonging to Cg. Then, Cd is a basis for ðY ; rÞ. For U 2 Cd , U ¼ \fFi : Fi 2 C; i 2 Kg where K is a finite set, By assumption and Proposition 3.12, we have f 1 ðU Þ ¼ \ff 1 ðFi Þ : i 2 Kg 2 LdGC ðX ; sÞ. h Remark 4.12. For a function f : ðX ; sÞ ! ðY ; rÞ, the following implications hold: LdC-continuous ! # sub-LdC-continuous !
LdGC -continuous # sub-LdGC -continuous
! !
GLC -continuous # sub-GLC -continuous
In the remark above, the relationships cannot be reversible as the following examples illustrate. Example 4.13. Let X ¼ Y ¼ fa; b; cg and s ¼ fX ; /; fag; fcg; fa; cg; fb; cgg and r be the topology induced by a base B of Y . Let f : ðX ; sÞ ! ðY ; rÞ be the identity function. (a) If B ¼ fY ; fag; fcgg, then f is sub-LdGC -continuous, but it is not dGLC -continuous, since f 1 ðfa; cgÞ 62 LdGC ðX ; sÞ for fa; cg 2 r. (b) If B ¼ fY ; fcgg, then f is sub-LdGC -continuous, but it is not sub-LdC-continuous, since f 1 ðfcgÞ ¼ fcg 2 LdGC ðX ; sÞ but f 1 ðfcgÞ 62 LdCðX ; sÞ. (c) If B ¼ fY ; fag; fcg; fa; cgg, then f is sub-GLC -continuous, but it is not sub-LdGC -continuous, since f 1 ðfa; cgÞ 2 GLC ðX ; sÞ but f 1 ðfa; cgÞ 62 LdGC ðX ; sÞ. The following result is an immediate consequence of Remark 3.5. Remark 4.14. The following statements hold for any f : ðX ; sÞ ! ðY ; rÞ: (a) If a space ðX ; sÞ is semi-regular, then sub-LdGC -continuous and sub-GLC -continuous coincide. (b) If a space ðX ; sÞ is T3=4 , then sub-LdGC-continuous and sub-LdC -continuous coincide. In conclusion, generalizations of closed sets in point-set topology suggest some new separation axioms which have been found to be very useful in the study of certain objects of digital topology. Thus we may stress once more the importance of locally d-generalized closed sets as a branch of them and the possible application in computer graphics [8,9] and quantum physics [4]. Acknowledgement This work is supported by grant no. R05-2002-000-01045-0 from the Korea Science & Engineering Foundation. References [1] Balachandran K, Sundaram P, Maki H. Generalized locally closed sets and GLC-continuous functions. Indian J Pure Appl Math 1996;27:235–44.
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