The kernel in fixed topological spaces

The goal of the article is to extend and study the proper fixed spaces. The sets called D-, and Kernel sets in this space have been studied and introduced. In this paper we include theorems and examples related to these concepts in fixed topological space. This research finds the characteristics and properties of such notions. Also the relation between them are studied and investigated in fixed topological spaces. Since the definition of the mathematics usual topology differs from the definition of fixed topology, we have noticed that some of the concepts that have been studied in both are different in terms of their verification in the proofs and examples and evidence for that the of definitions D-sets. α, R0, D0, D1 - spaces and the kernel set. Also we were obtained that the relationships between these concepts are different from those in previous studies.

sets -definitions D s and evidence for that the of proofs and example spaces and the kernel set. Also we were obtained that the relationships between these concepts are different from those in previous studies.

1.Introduction:
There is a definition derived from the definition of usual topology called a supra topological and it has been proposed by ( Mashhour et al. ,1983) [1] and it has also been studied in (Meera Devi et al. , 2016) [2].The fixed topological spaces were introduced, and studied in (Raad Aziz Hussain Alas space o defined R had [4] 1961) , Davis ( . ) s under the title (proper fixed space [3] ,2019 ) Abdulla if for every open set of G and for every x ∈ G, therefore {x} ̅̅̅̅ ⊆ G and this definitions were studied in (Bishwambhar Roy et al. ,2010) [5]. In addition, ( Munkres ,2000) [6] introduced the definition of the  Roy et al. ,2010) applied the definition of the kernel set, also this definition was studied in (Sindhu , 2019) [11] . The aim of writing this article is to explain these concepts of D-and kernel set in the fixed topological spaces, studying their characteristics and the relation through them. We used the(ft-s )symbol to denote the fixed topological spaces. Our study deepens the new concept called fixed technology, which differs in nature from the usual topology, as the two concepts are originally independent. In the second chapter of this work, we recalled the that had previously been discussed. As for the third chapter, we identified a new sets under the name D-sets. Then a new spaces were defined using the D-sets under the name α, D o and D 1 spaces . For fixed topological the relationships between these spaces are found and the possibility of equivalence between them are discuses. In the fourth chapter, we studied a new sets under the name kernel sets and we reached many important theories that belong to this sets related to fixed topological space, among them the behavior of the kernel sets in the presence of space and also the existence of a sets of intersections of all sets present in the fixed topological space.
Notation: We will use the(ft-s )symbol to denote the fixed topological spaces

On The F-open Sets:
In this chapter, we recalled the description of the F-open sets and its related theories and examples.

Definition 2.1 :[3]
Suppose X is a non-empty set and ℱ is a collection of sets on X fulfills the following axioms : ( i) If ∈ ℱ for all ∈ ∧ such that ∧ is any arbitrary set , thus ∪{ : ∈ ∧ } ≠ X.
Hence ℱ is named fixed topology on X and (X, ℱ) is named (ft-s   Thus ℱ is a fixed topological of X .If B ⊆ X ,then:    It is evident that in the case of ∈ ℱ, thus ∈ ( ).

On D-Sets:
In this chapter, we studied the definition of the D-set and the effect of this definition on spaces within the fixed topological space in terms of the different results obtained This in the following case :       Hence there is ∈ ℱ such as ∈ ∉ . Therefore by Theorem (3.19), ( , ℱ) is -space. It is obvious that the 1 -space is 0 -space. non F-open set F such as ∈ , ∈ ∈ , ∈ and ∩ = ∅. Therefore not every 0 -space  It is clear ( , ℱ) is 1 -space. But for 1 ≠ 2, there is no D-sets U,V such as 1 ∈ , 2 ∈ and ∩ = ∅.

On Kernel Sets:
In the fourth chapter, the definition of the kernel set has been included to obtain positive results, using the definitions and concepts that were discussed in the previous chapters.    Then  Then ∉ ̅ . Therefore ∉ ̅ ∩ ker( ). Then ̅ ∩ ker( ) ⊆ . Since ⊆ ̅ and ⊆ ker( ), then ⊆ ̅ ∩ ker( ). Therefore ̅ ∩ ker( ) = .   Proof : If A is D-set, then ∩ = ∅ for some ∈ ℱ. Therefore the proof follows from Theorems      Hence ker{ } ≠ ker{ } ∀ , ∈ ∋ ≠ , but ( , ℱ) is not 0 -space. The following example show that ker{ } = ker{ } for some , ∈ such that ≠ since the space is not 0 -space.

5.Results:
The most important results that were reached through this research are: The kernel of any set in the space is equal to the union of the set with the set of all intersection of open sets in the fixed topological space sets. If the kernel of two different points are equal, then the closure of the two points are equal too. In the 0 -space, the kernel of any two different points are not equal. Finally the kernel is distributed over the intersection and the union.

6.Conclusion:
In this research, we study the definition of (ft-s).This study introduced, examines and studied the Dset in (ft-s). To achieve the aim of the study , some properties and characterizations of these concepts are investigated. The kernel sets, that related to F-open sets are introduced and studied. We also obtained the fact that the relationships between these concepts are different from those in previous studies, and especially with our main concept, which is kernel set. This work will open a method for other researchers to study the applications of kernel sets.