165 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 G. S. Ashaea Y. Y. Yousif Abstract This paper consist some new generalizations of some definitions such: j-ω-closure converge to a point, j-ω-closure directed toward a set, almost j-ω-converges to a set, almost j-ω-cluster point, a set j-ω-H-closed relative, j-ω-closure continuous mappings, j-ω-weakly continuous mappings, j-ω-compact mappings, j-ω-rigid a set, almost j-ω-closed mappings and j-ω-perfect mappings. Also, we prove several results concerning it, where j {, δ,, pre, b, }. Keywords: Filter base, j-ω-closure converge, almost j-ω-converges, almost j-ω-cluster, j-ω- rigid a set, j-ω-perfect mappings. Math Subject Classification 2010: 54C05, 54C08, 54C10. 1. Introduction The notion "filter" first commence in Riesz [1]. and the setting of convergence in terms of filters sketched by Cartan in [2, 3]. And was sophisticatedly by Bourbaki in [4]. Whyburn in [5]. Introduces the notion directed toward a set and the generalization of this notion studied in Section 2. Dickman and Porter in [6]. Introduce the notion almost convergence, Porter and Thomas in [7]. introduce the notion of quasi-H-closed and the analogues of this notions are studied in Section 3. Levine in [8]. Introduce the notion θ-continuous functions, Andrew and Whittlesy in [9]. Introduce the notion weakly θ-continuous functions, in Dickman [6]. Introduce the notions θ-compact functions, θ-rigid a set, almost closed functions and the analogues of this notions are studied in Section 4. In [5]. The researcher introduces the notion of θ-perfect functions but the analogue of this notion studied in Section 5. The neighborhood denoted by nbd. The closure (resp. interior) of a subset K of a space G denoted by cl (K) (resp., int(K)). A point g in G is said to be condensation point of K ⊆ G if every S in τ with g  S, the set K ∩ S is uncountable [10]. In 1982 the ω-closed set was first exhibiting by Hdeib in [10]. and he know it a subset K ⊆ G is called ω-closed if it incorporates each its condensation points and the ω-open set is the complement of the ω-closed set [12]. The ω- interior of the set K ⊆ G defined as the union of all ω-open sets contain in K and is denoted by intω(K). A point g  G is said to θ-cluster points of K ⊆ G if cl(S) ∩ K ≠ φ for each open set S Ibn Al Haitham Journal for Pure and Applied Science Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index Doi:10.30526/32.3.2289 Filter Bases and j-ω-Perfect Mappings Department of Mathematics, College of Education for Pure Sciences, Ibn - Al-Haitham, University of Baghdad Dept. of Mathematics, College of Education for Pure Sciences, Ibn -Al- Haitham, University of Baghdad ghidaasadoon@gmail.com yoyayousif@yahoo.com Article history: Received 25 March 2019, Accepted 26 May 2019, Publish September 2019 mailto:ghidaasadoon@gmail.com mailto:yoyayousif@yahoo.com1 166 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 of G containment g. The set of each θ-cluster points of K is called the θ-closure of K and is denoted by clθ(K). A subset K ⊆ G is said to be θ-closed [11]. if K = clθ(K). The complement of θ-closed set said to be θ-open. A point g  G said to θ-ω-cluster points of K ⊆ G if ωclθ(S) ∩ K ≠ φ for each ω-open set S of G containment g. The set of each θ-ω-cluster points of K is called the θ-ω-closure of K and is denoted by ωclθ(K). A subset K ⊆ G is said to be θ-ω- closed [11]. if K = ωclθ(K). The complement of θ-ω-closed set said to be θ-ω-open, δ-closed [12]. if K= clδ(K) = {g  G: int(cl(S)) ∩ K ≠ φ, S  τ and g  S}. The complement of δ- closed said δ-open set, δ-ω-closed if K = ωclδ (K) = {g  G: intω(cl(S)) ∩ K ≠ φ, S  τ and g  S}. The complement of δ-ω-closed said δ-ω-open. 2. Filter In this section we introduce definition of filter, filter base, nbd filter, finer ultrafilter and some other related concepts. Definition 1 [4]. A nonempty family  of nonempty subsets of G called filter if it satisfies the following conditions: (a) If M1, M2  , then M1 ∩ M2  . (b) If M   and M  M*  G, then M* . Definition 2 [4]. A nonempty family  of nonempty subsets of G is called filter base if M1, M2   then M3  M1 ∩ M2 for some M3  . The filter generated by a filter base  consists of all supersets of elements of . An open filter base on a space G is a filter base with open members. The set g of all nbds of g  G is a filter on G, and any nbd base at g is a filter base for g. This filter called the nbd filter at g. Definition 3 [4]. Let  and be filter bases on G. Thenis called finer than  (written as  <) if for all M  , there is G  such that G  M and that  meets G if M ∩ G   for all M   and G . Notice,   g iff g < . Definition 4 [4]. A filter  is called an ultrafilter if there is no strictly finer filter than . The ultrafilter is the maximal filter. Definition 5 [13]. A subset K of a space G called: (a) -ω-open if K  intω(cl(intω(K))). (b) pre-ω-open if K  intω(cl(K)). (c) b-ω-open if K  cl(intω(K))  intω(cl(K)). (d) -ω-open if K  cl(intω(cl(K))). The complement of an (resp. -ω-open, pre-ω-open, b-ω-open, -ω-open) called (resp. - ω-closed, pre-ω-closed, b-ω-closed, -ω-closed). The j-ω-closure of K  G is denoted by cl j-ω-(K) and defined by cl j-ω-(K) = ∩{M  G; G is j-ω-closed and K  M}, where j{, δ,  , pre, b, }. Several characterizations of ω-closed sets were provided in [11],. [13-16]. Furthermore, we built some results about δ-ω-closed and δ-ω-open depending on the results in [17-19]. 167 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 3. Filter Bases and j-ω-Closure Directed toward a Set In this section we defined filter bases and j-ω-closure directed toward a set and the some theorems concerning of them. Lemma 6 [15]. Let  : (G, τ)  (H, σ) be an injective mapping. (a) If  = {M: M  G} is a filter base in G, then () = {(M): M  } is a filter base in H. (b) If  = {G: G  (G)} is a filter base in (G),  = { –1 (G): G  } is a filter base in G. For each   K  G and any filter base  in (K), then {K ∩  –1 (G): G  } is a filter base in K. (c) If  = {M : M  G} is a filter base in G,  = {(M) : M  }, G* is finer than G, and * = { –1 (G *) : G*  *}, then the collection of sets ** = {M ∩ M* for all M   and M*  *} is finer than both of  and *. Definition 7 [4]. Let  be a filter base on a space G. We say that  converges to g  G (written as   g) iff each open set S about g contains some element M  . We say  has g as a cluster point (or  cluster at g) iff each open set S about g meets all element M  . Clear that if   g, then  cluster at g. Definition 8 [15]. Let  be a filter base on a space G. We say that  directed toward (shortly, dir,- tow) a set K  G, provided each filter base finer than  has a cluster point in K. (Note: Any filter base can't be dir,- tow the empty set). Now, we will generalizations Definitions 7 and 8 as follows. Definition 9 Let  be a filter base on a space G. We say that  closure converges to g  G (written as  ⇝ g) iff all open set S about g, the cl(S) contains some element M  . We say  has g as a closure cluster point (or  closure cluster at g) iff all open set S about g the cl(S) meets all element M  . Clear that if ⇝g, then  closure cluster at g. cl (g) used to denote the filter base {cl(S): S  g}. Notice, ⇝g if and only if cl (g) < . [10]. Definition 10 Let  be a filter base on a space G. We say that  closure directed toward (shortly, cl dir,- tow) a set K  G, provided each filter base finer than  has a closure cluster point in K. Theorem 11 Let  be a filter base on a space G. ⇝g  G if and only if  is cl dir,- tow g. Proof: () Assume ⇝g, all open set S about g, cl(S) contains an element of  and thus contains an element of every filter base * < , therefore * actually closure converges to g. () Assume  is cl dir,- tow g, it must  ⇝ g. For if not, yond is an open set S in G about g such that cl(S) don't contains an element of . Denote by * the collection of sets M* = M ∩ (G  cl(S)) for M  , then the sets M* are nonempty. And * is a filter base and indeed * < , because result in M1* = M1 ∩ (G  cl(S)) and M2* = M2 ∩ (G  cl(S)), so there is an M3  M1 ∩ M2 and this perform to M3* = M3 ∩ (G  cl(S))  M1 ∩ M2 ∩ (G  cl(S)) = M1 ∩ (G  cl(S)) ∩ M2 ∩ (G  cl(S)). 168 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 By construction, g is not a closure cluster point of *. This contradiction crops that, ⇝g. Theorem 12 Let  : (G, τ)  (H, σ) be an injective mapping and given L  H. If for each filter base  in (G) cl dir,- tow a point h  L, the inverse filter M = { –1 (G) : G } is cl dir,- tow  – 1 (h), then for any filter base  in (G) cl dir,- tow a set L, E = {  –1 (M) : M  } is cl dir,- tow K =  –1 (L). Proof: Suppose that the hypothesis is true and any h  L is a closure cluster point of a filter base finer than  must be in (G). Thus L ∩ (G)  , and  is cl dir,- tow L ∩  (G). So we may assume L  (G). Let M be a filter base finer than E. Then  = {( (m): m M} finer than  by Lemma (6, a). So  has a closure cluster point l in L and a filter base * finer than  closure converges to l and so is cl dir,- tow l. By supposition M * = { –1 (G*): G **} is cl dir,- tow  –1 (l). In addition, by Lemma (6, c), M and M * have a common filter base M ** finer than of them. So M ** has a closure cluster point g in  –1 (l). Since g is a closure cluster point of M and g   –1 (l)  K, obtain result follows. Theorem 13 Let  : G  H be closed mapping and  –1 (h) compact for every h  H iff for every filter base  in (G) cl dir,- tow a set L  H, the collection E = { –1 (M) : M  } is cl dir,- tow  –1 (L). Proof: () Suppose that  is closed mapping and  –1 (h) compact for every h  H. Then by Theorem 11 and 12 it suffices to prove that if  is a filter base in  (G) j-ω-closure converging to h  L, then M = { –1 (G) : G } is cl-d-t  –1 (h).In order to if not, yond is a filter base M* finer than M, no point of  –1 (h) is a j-ω-closure cluster point of M *. For all g   –1 (h), by supposition yond is an open set Sg about g and M * g  M* with M * g ∩ Sg = . Since  –1 (h) is compact, yond are a finite numbers of open sets S ig such that  –1 (h)  S =  S ig , suppose m*  M* such that m*  ∩ m * ig and let T = H   (G  S) be the open set. Then (m*) ∩ T =  because of m*  G  cl(S). So since (m*)  *, * cannot have h as a closure cluster point. () Suppose that the hypothesis is true and  is not closed. Let K  G be a closed set and for some h  H  (K) is a closure cluster point of (K). Suppose  be a filter base of sets (K) ∩ T for every open sets T  H such that h  T, then is a filter base in  (G) and  ⇝ h. Let M = { –1 (G): G } and M* = {K ∩ m : m  M}. It apparent that M * < M. Nevertheless, G  K is open and  –1 (h)  G  K, M * has no closure cluster point in  –1 (h). The contradiction crops that  be a closed mapping. Finally, to prove  –1 (h) is compact, this is easy for h  H  (G). And for h  (G), {h} is a filter base in (G) cl dir,- tow h. By supposition, { –1 (h)} cl dir,- tow  –1 (h). This means that every filter base in  –1 (h) has a closure cluster point in  –1 (h), so that  –1 (h) is compact. Corollary 14 Let  : G  H be closed mapping and  –1 (h) compact for every h  H if and only if each filter base in (G) ⇝ h  H has pre-image filter base cl dir,- tow  –1 (h). 169 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 Corollary 15 Let  : G  H be closed mapping and  –1 (h) compact for every h  Y, for every compact set W  H,  –1 (W) is compact. Proof. Let W  H be a compact set and  is a filter base in  –1 (W),  = { (M): M  }, is a filter base in W and in  (G) and is cl dir,- tow W. So * = { –1 (G): G } is cl dir,- tow  – 1 (W), so that * <  and * has a closure cluster point in  –1 (W). 4. Filter Bases and Almost j-ω-Convergence In this section, we defined filter bases, almost j-ω-closure, and the some theorems about them. We now introduce the definition of almost j-ω-closure, where j {, δ, , pre, b, }. Definition 16 Let  be a filter base on a space G. We say  almost j-ω-converges to a subset K  G (written as j-ω ⇝K) if for each cover K of K by subsets open in G, there is a finite subfamily L  K and M   such that M  {cl (L) : L  L}. We say  almost j-ω- converges to g  G (written as  j-ω ⇝ g) if  j-ω ⇝ {g}. Now, cl (g) ⇝ g, while, j-ω cl (g) j-ω ⇝g, where j {, δ, , pre, b, }. Also, we introduce the definitions of almost j-ω-cluster point, and quasi -j-ω-H-closed set where j {, δ, , pre, b, }. Definition 17 A point g  G is called an almost j-ω-cluster point of a filter base  (written as g  (al- j- ω-cg)) if  meets cl j-ω-(g), where j {, δ, , pre, b, }. For a set K  G, the almost j-ω-closure of K, denoted as (al- j-ω-cl (K)) is al j-ω-cg {K} if K   i.e. {g  G: every j-ω-closed nbd of g meets K} and is  if K = ; K is almost j-ω-closed if K = (al- j-ω-cl(K)). Correspondingly, the almost j-ω-interior of K, denoted as (al- j-ω-intK), is {g  G; cl j-ω- (S)  K for some open set S containing g}; K is almost j-ω-interior if K = (al- j-ω-int(K)), where j{, δ, , pre, b, }. Theorem 18 Let  and  be filter bases on a space G, K  G and g  G. (a) If  j -ω ⇝ k, then cl j -ω (k) < . (b) If  j -ω ⇝ g, iff cl j -ω (g) < . (c) If  < , then (al- j -ω -cg)  (al- j -ω cg). (d) If  <  and  j -ω ⇝ K, then  j -ω ⇝ K. (e) (al- j -ω cg) = ∩ {cl j -ω (M): M  }. (f) If  j -ω⇝ g and g  K, then  j -ω ⇝K. (g) If  j -ω ⇝K iff  j -ω ⇝K ∩ (al- j -ω -cg). (h) If  j -ω ⇝K, then K ∩ (al- j -ω cg)  . (i) If S  G is open, then (al- j -ω -cl(S)) = cl(S). (j) If  is a open filter base, then (al- j -ω cl) = (al- j -ω cg). If S is an open ultrafilter on G. Then S ⇝g if and only if S j -ω ⇝g, where j {, δ, , pre, b,}. Proof: The proof is easy, so it omitted. 170 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 Definition 19 The subset K of a space G is said to be quasi -j-ω-H-closed relative to G if every cover K of K by open subsets of G contains a finite subfamily L  K such that K  {cl j-ω-(L) : L  B}. If G is Hausdorff, we say that K is j-ω-H-closed relative to G. If G is quasi- j-ω-H-closed relative to itself, then G is said to be quasi- j-ω-H-closed (resp. j-ω-H-closed), where j {, δ, , pre, b, }. Theorem 20 The following are equivalent for a subset K  G: (a) K is quasi-j-ω-H-closed relative to G. (b) For all filter base  on K, j-ω ⇝K. (c) For all filter base  on K, (al -j-ω cg) ∩ K  . Where j {, δ, , pre, , b, }. Proof: Clearly (a)  (b), and by Theorem (18, h), (b)  (c). To show (c)  (a), let K be a cover of K by open subsets of G such that the j-ω-closed of the union of any finite subfamily of K is not cover K. Then  = {K  cl j-ω-g(k Sk): k is finite subfamily of K} is a filter base on K and (al -j-ω-cg) ∩ K = . This contradiction crop s that K is quasi- j-ω-H-closed relative to G, where j {, δ, , pre, b, }. By concepts of closure directed toward a set, almost j-ω-convergence characterized and related in the next result. Theorem 21 Let  be a filter base on a space G and K  G . Then: (a)  is cl-dir,-tow K iff for each cover K of K by open subsets of G, there is a finite subfamily L  K and an M   such that M  {cl j-ω-(L) : L B }, where j {, δ, , pre, , b, }. (b) For every filter base,  <  implies (al- j-ω-cg) ∩ K   iff j –ω ⇝ K, where j  {, δ, , pre, b, }. Proof: The proofs of the two facts are similar; so, we will only prove the fact (b): () Suppose for every filter base,  <  implies (al- j-ω-cg) ∩ K  . If j –ω ⇝ g for some g  K, then by Theorem (3.3, f), j-ω ⇝K. So, assume that for each g  K,  does not j-ω ⇝g. Let K be a cover of K by subsets open in G. For every g  K, yond is an open set Sg containing g and Tg  K such that Sg  Tg and M cl j-ω-g(Sg)   for every M  . So, g = {M  cl j-ω-g(Sg) : M  } is a filter base on G and  < g . Now, g  (al- j-ω-cgg). Assume that {g: g  K} forms a filter sub base with  denoting the generated filter. Then  <  and (al -j-ω-cg) ∩ K = . This contradiction implies yond is a finite subset L  K and Mg   for g  L such that,  = ∩{Mg  cl j-ω- g(Sg) : g  L}.There is M   such that M  ∩{Mg : g  L}. It easily follows that  = ∩{M  cl j-ω-g(Sg) : g  L and M  {cl j-ω- g(Tg): g  L}. Thus j-ω ⇝K. () Suppose j-ω ⇝K and  is a filter base such that  < . By Theorem (18, d), j-ω ⇝ K, and Theorem (18, h), (al- j-ω-cg) ∩ K  . 171 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 5. Filter Bases and j-ω-Rigidity In the section, we defined filter bases, j-ω-rigidity, and the some theorems concerning of them. Definition 22 A mapping  : G  H is said to be j-ω-closure continuous (resp. j-ω-weakly continuous) if for every g  G and every nbd T of  (g), there exists a nbd S of g in G such that (cl j-ω- (S))  cl j-ω-(T) (resp. (S)  cl j-ω-(T)). Clearly, every continuous mapping is j-ω-closure continuous, where j{, δ, , pre, b, }. The notions of almost j-ω-convergence and almost j-ω-cluster can used to characterize j-ω- closure continuous. Theorem 23 Let  : G  H be a mapping. The following are equivalent: (a)  is j-ω-closure continuous. (b) For all filter base  on G, j-ω ⇝g implies  ()   (g). For all filter base  on G,  (al- j-ω-c)  (al- j-ω-c  (). For all open S  H,  -1 (S)  (al- j- ω-int –1 (al- j-ω-cl(S))). Where j { , δ, , pre, b, }. Proof: The proof of the equivalence of (a), (b) and (d) is straightforward. (a)  (c) Suppose  is a filter base on G, g  (al- j-ω-c), M   and T is a nbd of  (g), yond is a nbd S of g such that  (cl j-ω- (S))  cl j-ω-(T). Since cl j-ω-(S) ∩ M  , then cl j- ω-(T) ∩  (M)  . So,  (g)  (al- j-ω-c ()). This shows that (al- j-ω-c)  (al- j-ω-c ()). (c)  (a) Let S be an ultrafilter containing (cl j-ω-(g)). Now,  –1 (S) is a filter base since (G)  S and  –1 (S) meets cl j-ω-(g). So,  –1 (S)  cl j-ω-(g) is contained in some ultrafilter T. Now  –1 (S) is an ultrafilter base that generates S. Since  –1 (S) < (T), then (T) also generates S; hence (al- j-ω-c(T )) = (al- j-ω-c S ). Since g  (al- j-ω-c(T )), then (g)  (al- j-ω-c T )  (al- j-ω-c (T )) = (al- j-ω-c S ). So, S meets cl j-ω- ((g)) and cl j-ω- ((g))  ∩{ S : S ultrafilter, S  (cl j-ω-(g))}, (denote this intersection by ). Nevertheless,  is the filter generated by (cl j-ω-(g)) (see [4]. Proposition I.6.6), so clj-ω ((g)) < (cl j-ω (g)). Hence  is j-ω-closure continuous, where j{, δ, , pre, b, }. Corollary 24 If : G  H is j-ω-closure continuous and K  G, then (al- j-ω-cl(K))  (al- j-ω- cl((K)), where j{, δ, , pre, b, }. Here are some similarly proven facts about j-ω-weakly continuous mapping. Theorem 25 Let  : G  H be a mapping. The following are equivalent: (a)  is j-ω-weakly continuous. (b) For all filter base  on G,   g implies () j-ω ⇝ (g). (c) For all filter base  on G, (al- j-ω-c)  (al- j-ω-c  ()). (d) For all open S  H,  –1 (S)  int  –1 (cl j-ω-(S)). Where j {, δ, , pre, b, }. 172 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 Theorem 26 If  : G  H is j-ω-weakly continuous mapping, then (a) For all K  G, (cl j-ω-(K))  (al- j-ω-cl (K)). (b) For all L  H, (cl j-ω-(int(cl j-ω- –1 (L))))  cl j-ω-(L). (c) For all open S  H,  (cl j-ω-(S))  cl j-ω-(S). Where j  {, δ, , pre, b, }. Now, We introduce the definitions of j-ω-compact, j-ω-rigid set, almost j-ω-closed, and j- ω-urysohn space as follows. Definition 27 A mapping  : G  H is said to be j-ω- compact if for every subset C quasi- j-ω-H-closed relative to H,  –1 (C) is quasi- j-ω-H-closed relative to G, where j  {, δ, , pre, b, }. Definition 28 A subset K of a space G is said to be j-ω-rigid provided whenever  is a filter base on G and K ∩ (al- j-ω-cg ) = , there is an open S containing K and M   such that cl j-ω-(S) ∩ M = , where j {, δ, , pre, b, }. Definition 29 A mapping  : G  H is said to be almost j-ω-closed if for any set K  G , (al- j-ω-cl(K)) = (al- j-ω-cl (K)), where j {, δ, , pre, b, }. Definition 30 A space G is said to be j-ω-Urysohn if every pair of distinct points are contained in disjoint j-ω-closed nbds, where j {, δ, , pre, b, }. Before characterizing j-ω-rigidity, we can show that a j-ω-closure continuous, j-ω-compact mapping into a j-ω-Urysohn space with a certain property (the “j-ω-closure” and “quasi- j-ω- H-closed relative” analogue of property  in [15].) is almost j-ω-closed. Theorem 31 Suppose  : G  H is a j-ω-closure continuous mapping and j-ω-compact and H is j-ω- Urysohn with this property: For each L  H and h  (al- j-ω-cl(L), there is a subset C quasi-j- ω-H-closed relative to H such that h  (al- j-ω-cl(C ∩ L)). Then  is almost j-ω-closed, where j {, δ, , pre, b, }. Proof: Let K  H. By corollary (24),  (al- j-ω-cl (K))  (al- j-ω-cl (K)). Suppose h  (al- j- ω-cl (K)).Yond is a subset C quasi- j-ω-H-closed relative to H such that h (al- j-ω-cl(C ∩ (K)). Then  = {cl j-ω-(S) ∩ C ∩ (K): S  h}, is a filter base on H such that  j-ω ⇝h. Now,  = {K ∩  –1 (M): M  } is a filter base on K ∩  –1 (C). Since  –1 (C) is quasi- j-ω-H- closed relative to H, then there is g  (al -j-ω-cg) ∩  –1 (C). By theorem 23, (g)  (al- j-ω- ch())  (al- j-ω ch). Since  j-ω ⇝ h and H is j-ω-Urysohn, (al- j-ω-ch) = {h}. So, h  (al- j-ω-cl (K)), where j {, δ, , pre, b, }. Theorem 32 Let K be a subset of a space G. The following are equivalent: (a) K is j-ω-rigid in G. 173 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 (b) For all filter base  on G, if K ∩ (al- j-ω-cg) = , then for some M  , K ∩ (al- j-ω- cl(M) = . (c) For all cover K of K by open subsets of G, there is a finite subfamily B  K such that K  int cl j-ω-( B). Where j {, δ, , pre, b, }. Proof: The proof that (a)  (b) is straightforward. (b)  (c) Let K be a cover of K by open subsets of G and  = {∩SB (G  cl j-ω-(S)): B is a finite subset of K}. If  is not a filter base, then for some finite subfamily B  K , G  {cl j-ω-(S) : S  B }; thus, K  G  int cl j-ω- ( B ) which completes the proof in the case that  is not a filter base. So, suppose  is a filter base. Then K ∩ (al- j-ω-c) =  and there is an M   such that K ∩ (al- j-ω-cl (M)) = . For each x  K, yond is open Tg of g such that cl j-ω-(Tg) ∩ M = . Let T = {Tg : g  K}. Now, T ∩ M = . Since M  , then for some finite subfamily B  K , M = ∩{G  cl j-ω-(S) : S  B }. It follows that T  cl j-ω- (B) and hence, K  int cl j-ω-( B), where j {, δ, , pre, b, }. (c)  (a) Let  be a filter base on G such that K ∩ (al -j-ω-c) = . For all g  K yond is open Tg of g and Mg   such that cl j-ω-(Tg) ∩ Mg = . Now {Tg: g  K} is a cover of K by open subsets of G; so, there is finite subset L  K such that K  int cl j-ω- ({Tg: g  T}). Let S = int cl j-ω-({Tg: g  L}). Yond is M   such that M  ∩ {Mg: g  L}. Since cl j-ω- (S) = {cl j-ω-(Tg): g  L}, then cl j-ω-(S) ∩ M = . So K is j-ω-rigid in G, where j {, δ, , pre, b, }. 6. Filter Bases and j-ω-Perfect Mappings In the section, we defined filter bases, j-ω-perfect mappings, and the some theorems about them. In Corollary 14, we show that a mapping  : G  H is perfect (i.e. closed and  –1 (y) compact for each h  H) iff for all filter base  on (G), ⇝h  H, implies  –1 () is (cl-dir- tow)  –1 (y) and in Corollary 15, proved that a perfect mapping is compact (i.e. inverse image of compact sets are compact). In view Theorem 21, we say that a mapping  : G  H is j-ω- perfect if for every filter base  on (G),  j-ω ⇝ h  H implies  –1 ()j-ω ⇝ –1 (h), where j {, δ, , pre, b, }. Theorem 33 Let  : G  H be a mapping. The following are equivalent: (a)  is j-ω-perfect. (b) For all filter base  on G, (al- j-ω-(c ())  (al- j-ω-(c). (c) For all filter base  on (G),  j-ω ⇝L  H, implies  –1 () j-ω ⇝  –1 (L). Where j  {, δ, , pre, b, }. Proof: (a)  (b) Assume  is a filter base on G and h  (al- j-ω-c ()). For if not. Assume that  –1 (h) ∩ (al- j-ω-(c) = . For each g   –1 (h), yond is open Sg of g and Mg   such that cl j-ω-(Sg) ∩ Mg = . Since  –1 (cl j-ω-(h)) j-ω ⇝ –1 (y) and {Sg : g   –1 (h)} is an open cover of  –1 (y), yond is a V  h and a finite subset B   –1 (y) such that  –1 (cl j-ω-(T))  {cl j-ω-(Tg): g  L }. Yond is an M   such that M  ∩ {Mg: g  L}. Thus, M ∩  –1 (cl j- 174 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 ω-(T)) =  implying cl j-ω-(T) ∩ (M) = , a contradiction as h  (al- j-ω-c ()). This shows that h  (al- j-ω-c), Where j{pre, , b, , }. (b)  (c) Assume  is a filter base on (G) and  j-ω ⇝L  H. Let be a filter base on G such that  –1 () < . Then  < () and (al- j-ω c ()) ∩ L  . Therefore (al- j-ω-c) ∩ L   and (al- j-ω-c) ∩  –1 (L)  . By Theorem (3.6, b),  –1 () j-ω ⇝ –1 (L), where j {, δ, , pre, b, }. (c)  (a) Clearly. Corollary 34 If  : G  H is j-ω-perfect mapping, then: (a) For all K  G, (al- j-ω-cl (K))  (al- j-ω-cl (K)). (b) For all almost j-ω-closed K  G, (K) is almost j-ω-closed. (c)  is j-ω-compact. Where j {, δ, , pre, b, }. Proof: (a) is an immediate consequence of Theorem 33, and (b) follows easily from (a). To prove (c) Let C be quasi- j-ω-H-closed relative to H , and  be a filter base on  –1 (C), then () is a filter base on C. By Theorem 20, (al- j-ω-c()) ∩ C   and by Theorem (33, b), (al- j-ω-c) ∩  –1 (C)  . By Theorem 20,  –1 (C) is quasi- j-ω-H-closed relative to G, where j{, δ, , pre, b, }. Theorem 35 An j-ω-closure continuous mapping  : G  H is j-ω-perfect if and only if (a)  is almost j-ω-closed, and (b)  –1 (y) j-ω-rigid for each h  H, where j {, δ, , pre, b, }. Proof: () If  is j-ω-closure continuous and j-ω-perfect mapping, then by Corollaries 34 and 24,  is almost j-ω-closed. To show  –1 (h), for h  H, is j-ω-rigid, Let  be a filter base on G such that  –1 (h) ∩ (al -j-ω-c) = . So, h  (al- j-ω-c) and by Theorem (33, b), h  (al j-ω c ()). Yond is open S of h and M   such that cl j-ω-(S) ∩ (M) = . So,  –1 (cl j-ω- (S)) ∩ M = . Since  is j-ω-closure continuous, then for any g   –1 (h), yond is open T of g such that cl j-ω-(T)   –1 (cl j-ω-(S)). So,  –1 (h) ∩ cl j-ω-(M) = , where j {, δ, , pre, b, }. () Assume that j-ω-closure continuous mapping  satisfies (a) and (b). Let  be a filter base on (G) such that  j-ω ⇝h. Let  be a filter base on G such that  –1 () < . So,  < () implying that h  (al- j-ω-c ()). Therefore, for each G  , h  (al- j-ω-cl( G ))  (al- j-ω-cl G ). Hence,  –1 (h) ∩ (al- j-ω-cl G)   for each G . By (b),  –1 (h) ∩ (al- j-ω- c)  . By Theorem 33,  is j-ω-perfect mapping, where j  {, δ, , pre, b, }. Actually, in the proof of the converse of Theorem 35, we have shown that property (a) of Theorem 35 can reduced to this statement: For each K  G, al j-ω-cl (K)   (al j-ω-cl (K); in fact, we have shown the next corollary (the mapping is not necessarily j-ω-closure continuous). Corollary 36 Let  : G  H be a mapping if (a) For all K  G, (al- j-ω-cl (K))  (al- j-ω-cl (K) 175 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 (b)  –1 (h) j-ω-rigid for each h  H, then  is j-ω-perfect, where j {, δ, , pre, b, }. Corollary 37 Let  : G  H be a mapping. (a)  is almost j-ω closed (b)  –1 (h) j-ω rigid for each h  H, then  –1 preserves j-ω rigidity, where j{, δ, , pre, b, }. Proof. Let C  H be j-ω rigid and  be a filter base on G such that al j-ω cg ∩  –1 (C) = . By Corollary 36 and Theorem 33, (al- j-ω c()) ∩ C = . So, there is M   such that (al- j- ω cl (M)) ∩ C = . Nevertheless (al- j-ω cl (M)) = (al- j-ω cl (M)). So, (al- j-ω cl (M)) ∩  –1 (C) = . So, by Theorem 32,  –1 (C) is j-ω rigid, where j {, δ, , pre, b, }. Theorem 38 Suppose  : G  H has j-ω rigid point-inverses. Then: (a)  is j-ω closure continuous iff for each h  H and open set T containing h, there is an open set S containing  –1 (h) such that (cl j-ω (S))  cl j-ω(T), where j {, δ, , pre, b, }. (b) If for each h  G and open set S containing  –1 (h), there is an open set T of h such that  – 1 (cl j-ω (T))  cl j-ω (S), then for each K  G , (al- j-ω cl((K))  (al- j-ω cl(K)), where j{, δ, , pre, b, }. Proof. (a) () Is obvious. () Is straightforward using Theorem (32, c) (b) Let   K  G and h  (al- j-ω cl (K)). Then  –1 (h) ∩ (al- j-ω cl (K)) = . Now,  = {K} is a filter base and (al -j-ω c) ∩  –1 (h) = . So, yond is open set S continuing  –1 (h) such that cl j-ω (S) ∩ K = , yond is open T of h such that  –1 (cl j-ω(T))  cl j-ω(S). Therefore, cl j-ω (T) ∩  (K) = . Hence h  (al- j-ω cl (K)), where j{, δ, , pre, b, }. The next result related to Theorem (38, b); the proof is straightforward. Theorem 39 Let  : G  H. The following are equivalent: (a) For all j-ω-closed K  G, (K) is j-ω-closed, where j {, δ, , pre, b, }. (b) For all L  H and j-ω open S containing  –1 (L), there is j-ω-open T containing L such that  –1 (T)  S, where j {, δ, , pre, b, }. Theorem 40 If  : G  H is j-ω closure continuous and H is j-ω Urysohn, then  is j-ω perfect if and only if for all filter base  on G, if () j-ω ⇝h  H, then (al- j-ω cg)  , where j{, δ, , pre, b, }. Proof. () Assume that  is j-ω perfect and () j-ω ⇝h. Therefore,  –1 () j-ω ⇝ –1 (h). Since  –1 () < , then by Theorem ( 18, d),  j-ω ⇝ –1 (h), by Theorem (18, h), (al - j-ω c )  . () Assume that for each filter base  on G, if () j-ω ⇝h  G, then (al- j-ω cg) . Suppose  is a filter base on (G) such that  j-ω ⇝h  H, and assume L is a filter base on G such that  –1 () < L. Then  =  –1 (G) <  (L). So, (L) j-ω ⇝ h. Therefore, (al- j-ω-cg 176 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 L)  . Let i  H  {h}. Because of H j-ω-Urysohn, yond are open sets Si of i and Sh of h such that cl j-ω-(Si) ∩ cl j-ω-(Sh) = . Yond is H  L such that (H)  cl j-ω- (Sh). For every g   –1 (i), there is open Ti of i such that  (cl j-ω- (Ti))  cl j-ω- (Si). So, cl j-ω- (Tg) ∩ H = . It follows that  –1 (i) ∩ (al- j-ω-cg L) =  for each i  H  {h}. So, (al- j-ω-cg L) ∩  –1 (h)   and  is j-ω-perfect, where j{, δ, , pre, b, }. Corollary 41 If  : G  H be a mapping is j-ω-closure continuous, G is quasi- j-ω-H-closed, and H is j-ω-Urysohn, then  is j-ω-perfect, where j{, δ, , pre, b, }. Proof. Since G is quasi- j-ω-H-closed, then all filter base on G has non void almost j-ω- cluster; now, the corollary follows directly from Theorem 35, Where j{, δ, , pre, , b, }. 7. Conclusions The starting point for the application of abstract topological structures in j-ω-perfect mapping is presented in this paper. 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