Filter Bases and j-ω-Perfect Mappings

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, }.


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 ωinterior 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 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-ωclosed [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-

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: 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 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].

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].
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.
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 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 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 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 () 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 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).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).

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-ωconverges 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-jω-c g )) if  meets cl j-ω-( g ), where j {, δ, , pre, b, }.

Theorem 18
Let  and  be filter bases on a space G, K  G and g  G.
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.

Definition 19
The subset K of a space G is said to be quasi

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.
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-ωg ( k S k ): k is finite subfamily of K} is a filter base on K and (al -j-ω-c g ) ∩ 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, }.

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-c g ) ∩ 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 S g containing g and T g  K such that S g  T g and M cl j-ωg (S g )   for every M  .So,  g = {M  cl j-ωg (S g ) : M  } is a filter base on G and  <  g .Now, g  (al-j-ω-c g  g ).
Assume that { g : g  K} forms a filter sub base with  denoting the generated filter.Then  <  and (al -j-ω-c g ) ∩ K = .This contradiction implies yond is a finite subset L  K and M g   for g  L such that,  = ∩{M g  cl j-ωg (S g ) : g  L}.There is M   such that M  ∩{M g : g  L}.It easily follows that  = ∩{M  cl j-ωg (S g ) : g  L and M  {cl j-ωg (T g ): 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-c g ) ∩ K  .

Filter Bases and j-ω-Rigidity
In the section, we defined filter bases, j-ω-rigidity, and the some theorems concerning of them.
The notions of almost j-ω-convergence and almost j-ω-cluster can used to characterize j-ωclosure continuous.

Theorem 23
Let  : G  H be a mapping.The following are equivalent: (a)  is j-ω-closure continuous.

Proof:
The proof of the equivalence of (a), (b) and (d) is straightforward.
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.

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 ∩ (alj-ω-c g  ) = , there is an open S containing K and M   such that cl j-ω-(S) ∩ M = , 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, }.

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- , 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. (c)  (a) Let  be a filter base on G such that K ∩ (al -j-ω-c) = .For all g  K yond is open T g of g and M g   such that cl j-ω-(T g ) ∩ M g = .Now {T g : g  K} is a cover of K by open subsets of G; so, there is finite subset

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-dirtow)  -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-ωperfect 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: . This shows that h  (alj-ω-c), Where j{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).
Proof.Since G is quasi-j-ω-H-closed, then all filter base on G has non void almost j-ωcluster; now, the corollary follows directly from Theorem 35, Where j{, δ, , pre, , b, }.

Conclusions
The starting point for the application of abstract topological structures in j-ω-perfect mapping is presented in this paper.We use filter base to introduce a new notion namely filter base and j-ω-perfect mapping.Finally, certain theorems and generalization concerning these concepts of studied; j {, δ, , pre, b, }.
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  by supposition yond is an open set S g about g and M * g  M* with M * g ∩ S g = .Since  -1 (h) is compact, yond are a finite numbers of open sets S i g such that  -1 (h)  S =  S i g , suppose m*  M* such that m*  ∩ m * i g 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.
Ibn Al-Haitham Jour.for Pure & Appl.Sci.32 (3) 2019 Corollary 15 Al-Haitham Jour.for Pure & Appl.Sci.32 (3) 2019 (b) For all filter base  on G, if K ∩ (alj-ω-c g ) = , then for some M  , K ∩ (alj-ωcl(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 ∩ (alj-ω-c) =  and there is anM   such that K ∩ (alj-ω-cl (M)) = .For each x  K, yond is open T g of g such that cl j-ω-(T g ) ∩ M = .Let T = {T g : 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, }.