Results on a PreT 2 Space and Pre-Stability

This paper contains an equivalent statements of a pre-T2 space, where  = {(x, x) x  X} and K = {(x1, x2) XX  f(x1) = f(x2)} are considered subsets of XX with the product topology. An equivalence relation between the preclosed set  and a pre-T2 space, and a relation between a pre-T2 space and the preclosed set K with some conditions on a function f are found. In addition, we have proved that the graph C of R is preclosed in XX, if X/R is a pre-T2 space, where the equivalence relation R on X is open. On the other hand, we introduce the definition of a pre-stable (γpre-stable) set by depending on the concept of a pre-neighborhood, where we get that every stable set is pre-stable. Moreover, we obtain that a pre-stable (γpre-stable) set is positively invariant (invariant), and we add a condition on this set to prove the converse. Finally, a relationship between, (i) a pre-stable (γpre-stable) set and its component (ii) a preT2 space and a (positively critical point) critical point, are gotten.


Introduction:
This paper consists of three sections, section one is called "Introduction", where the contents of this paper are explained and fundamental concepts are given.The concept of a preopen set lies in (1), where they have introduced "a preopen set  when     and its properties such as the intersection of an open set and a preopen set is preopen".Also, "the union of any family of preopen sets is a preopen set", you can find it in (2).While "its complement (i.e.preclosed)" you see it in (1).References (1) and (3) show that, "the intersection of all preclosed sets containing A is called the preclosure of , denoted by   , which is the smallest preclosed set containing ".But (4) and (5) introduce a preneighborhood and a preopen function respectively.On the other side, many discussed preopen sets and their relationship with other sets, which explain in (7).By occasion, the concepts of "invariant, positively invariant and a stable set", you find it in (8).Besides, stability plays a significant role in various areas of life, for example, in the field of pharmacy as in (9), in engineering (10) and (11).In addition to mathematics (12), and (13), besides, in chemistry (14).
Mustansiriyah University, College of Science, Mathematics Department, Baghdad, Iraq E-mail: dr.anmar@uomustansiriyah.edu.iqMoreover, section two has the name "A Relation Between a pre- 2 Space and Some Sets", where we introduce an equivalent statements of a pre- 2 space.
In this section we study the relation between a pre- 2 space and the graph  of , and the sets  = {(x, x) x  X},  = {(x 1 , x 2 ) XX  f(x 1 ) = f(x 2 )}, which are considered subsets of XX with the product topology.On the other side, an equivalence relation between the preclosed set  and a pre-T 2 space, and a relation between a pre-T 2 space and the preclosed set  with some conditions on a function  are found.In addition, we have proved that the graph Section three is called "A Pre-Stable Set ", where we introduce the definition of a pre-stable (pre-stable) set by depending on the concept of pre-neighborhood.We prove theorems on this set, which illustrate their characteristics.For example, we get that every stable set is pre-stable.Moreover, we have obtained that a pre-stable (pre-stable) set is positively invariant (invariant), and we add a condition on this set to prove the converse.Finally, the relationships between, (i) a pre-stable (prestable) set and its component (ii) a pre-T 2 space and a (positively critical point) critical point, are gotten.

Open Access
Now we recall the following theorems and definitions that we need: Theorem 1.1.(2).A topological space  is called pre- 2 if and only if for each x, y  X and x  y , there exist two preopen sets  and  such that   ,   and  ∩  = .Theorem 1.2.(3).Let (X i ) i∈I be a family of topological spaces and      X i for each I.

Relation between a Pre-T 2 Space and Some Sets:
This section is devoted to discuss a relation between a pre- 2 space and some sets, which is defined on a product topological space.Besides, we include theorems that we have proved in the following, where  = {(x, x) x  X} and  = {(x 1 , x 2 ) XX  f(x 1 ) = f(x 2 )} are considered subsets of XX with the product topology.
B which leads to a contradiction.Hence by 2.1, we get that  is a preclosed subset of  .
 Let  be a preclosed subset of .
Let ,   with   .Then (, )  Since  is a preclosed subset of  , then by 2.1  a preopen subset  of   s.t.(, )  and Since  is a preopen set , then Let      =   Let  be the set which contains (, ) and ( ) Suppose that  .
If   and since    ̅ °, we get that  ∩   ∅ Hence, (  ) ∩ ∆   .Which leads to a contradiction By the same way  .Now either  ,   .So there are preopen sets   and   in  with    ,     ∪   and  ∪   are preopen subsets of  which contains x and y respectively .By 1.2 , we have Or   ,   .There exists a preopen set   which contains y.  ∪ V  is a preopen subset of  and  ∩ ( ∪ V 1 ) = ∅ By the same way if  ,  .If ,  , then  and  are preopen sets ,and  ∩  = ∅ So  is pre- 2 space.Theorem 2.4.Let  be a preirresolute function from a topological space  to a pre- 2 space .Then the set  is preclosed.
Since  is pre- 2 , then there exist  and  preopen subsets of  such that ()  , ()   , and Since  is preirresolute, then  −1 () and  −1 () are preopen subsets of .So by 1.2 , we get that Suppose that and ( 1 ) , ( 2 )  . that is  ∩   ∅ , which leads to a contradiction.So,  is a preclosed subset of    .Theorem 2.5.Let  be a homeomorphism function from a topological space  to a topological space  such that the set  is preclosed.Then  is a pre- 2 space.
is preopen in   for each I and   is a non dense for only finitely many I.A function from a topological space  into a topological space  is called: (i) Preirresolute if and only if the inverse image of any preopen set in  is a preopen set in .(ii)Preirresolute if and only if the inverse image of any preclosed set in  is a preclosed set in .A function from a topological space  into a topological space  is called almost preopen if and only if the direct image of any preopen set in  is also a preopen set in .
Definition 1.4.(2).Definition 1.6.(8).(i)A set  ⊂  is called positively invariant, if  ∈  ∀ ∈ , ∀ ∈  + .(ii) A set  ⊂  is called invariant, if  ∈  ∀ ∈ , ∀ ∈ .(iii) A point  ∈  is called a critical point, if  =  ∀ ∈ Theorem 2.1.Let  be a topological space.A subset  of  is preclosed if and only if     a preopen subset  of  ,which contains  and  ∩  = ∅ .Proof: Clear.Theorem 2.2.A topological space  is pre- 2 if and only if the set  is preclosed.Proof:  Let  be a pre- 2 space and (, )  , that is .PO(XX) s. t. xA , yB and A ∩ B = ∅.So A  B PO(XX) and (x, y) if  1   and  2 .If  1   and  2  , then () and () are preopen subsets of  which contain  1 and  2 respectively and () ∩ () = ∅.Hence  is a pre- 2 space.Let f be a homeomorphism function from a topological space X in to a topological space Y. Then Y is a pre-T 2 space if and only if the set K is preclosed.Proof: Clear Theorem 2.7.Let  be a topological space and  be an equivalence relation on  where  is open .Then the graph  of  is preclosed in   if / is a pre- 2 space.