On iα-Open Sets

219 On iα Open Sets Amir A. Mohammed Omar y. kahtab College of Education, University of Mosul Received on: 22/04/2012 Accepted on: 28/06/2012 ABSTRACT In this paper, we introduce a new class of open sets defined as follows: A subset A of a topological space ) , (  X is called iα-open set, if there exists a non-empty subset O of X, ) (X O O   , such that ) ( O A Cl A   . Also, we present the notion of iα-continuous mapping,


Introduction and Preliminaries
A Generalization of the concept of open sets is now well-known important notions in topology and its applications. Levine [7] introduced semi-open set and semicontinuous function, Njastad [8] introduced α-open set, Askander [15] introduced iopen set, i-irresolute mapping and i-homeomorphism, Biswas [6] introduced semi-open functions, Mashhour, Hasanein, and El-Deeb [1] introduced α-continuous and α-open mappings, Noiri [16] introduced totally (perfectly) continuous function, Crossley [11] introduced irresolute function, Maheshwari [14] introduced α-irresolute mapping, Beceren [17] introduced semi α-irresolute functions, Donchev [4] introduced contra continuous functions, Donchev and Noiri [5] introduced contra semi continuous functions, Jafari and Noiri [12] introduced Contra-α-continuous functions, Ekici and Caldas [3] introduced clopen-T1, Staum [10] introduced, ultra hausdorff, ultra normal, clopen regular and clopen normal, Ellis [9] introduced ultra regular, Maheshwari [13] introduced s-normal space, Arhangel [2] introduced α-normal space. The main aim of this paper is to introduce and study a new class of open sets which is called iα-open set and we present the notion of iα-continuous mapping, iα-totally continuity mapping and some weak separation axioms for iα-open sets. Furthermore, we investigate some properties of these mappings. In section 2, we define iα-open set, and we investigate the relationship with, open set, semi-open set, α-open set and i-open set. In section 3, we present the notion of iα-continuous mapping, iα-open mapping, iα-irresolute mapping and iα-homeomorphism mapping, and we investigate the relationship between iαcontinuous mapping with some types of continuous mappings, the relationship between iα-open mapping, with some types of open mappings and the relationship between iαirresolute mapping with some types of irresolute mappings. Further, we compare iαhomeomorphism with i-homeomorphism. In section 4, we introduce new class of mappings called iα-totally continuous mapping and we introduce i-contra-continuous mapping and iα-contra-continuous mapping. Further, we study some of their basic properties. Finally in section 5, we introduce new weak of separation axioms for iαopen set and we conclude iα-continuous mappings related with iα-separation axioms. Throughout this paper, we denote the topology spaces ) , simply by X and Y respectively. We recall the following definitions, notations and characterizations. The closure (resp. interior) of a subset A of a topological space X is denoted by ).  is said to be i-homeomorphism [15] if f is an i-continuous and i-open. [15].

Lemma 1.6
Every semi-open set in a topological space is an i-open set [15]. [15].

Sets That are iα-Open Sets and Some Relations With Other Important Sets
In

Lemma 2.5 Every i-open set in any topological space is an iα-open set.
Proof. Let X be any topological space and A  X be any i-open set. Therefore,

Mappings That are iα-Continuous and iα-Homeomorphism
In this section, we present the notion of iα-continuous mapping, iα-irresolute mapping and iα-homeomorphism mapping. is an iαcontinuous. is an iα-open.

Proposition 3.8 Every i-open mapping is an iα-open.
Proof.
be an i-irresolute mapping and V be any iα-open set in Y. Since, Remark 3.14 The following example shows that iα-irresolute mapping need not be irresolute, semi α-irresolute, α-irresolute and i-irresolute mappings.
is an iαhomomorphism.
Proof. Since, every i-continuous mapping is an iα-continuous by proposition 3.3. Also, since every i-open mapping is an iα-open 3.8. Further, since f is bijective. Therefore, f is an iα-homomorphism ■ The converse of the above theorem need not be true as shown in the following example

Mappings That are iα-Totally Continuous and iα-Contra-Continuous
In this section, we introduce new classes of mappings called iα-totally continuous, i-contra-continuous and iα-contra-continuous. continuous, but f is not iα-totally continuous because for iα-open set {a,c}, f -1 {a,c}={a,c} CO(X).

Proof. Let
be iα-totally continuous and V be an iα-open set in Y. Since, f is an iα-totally continuous mapping, then f -1 (V) is clopen set in X, which implies f -1 (V) open, it follow f -1 (V) iα-open set in X. Therefore, f is an iα-irresolute■ The converse of the above theorem need not be true as shown in the following example is an iα-irresolute, but f is not iα-totally continuous because for iα-open subset{1,3}, f -1 {1,3}={1,3} CO(X).

Theorem 4.7
The composition of two iα-totally continuous mapping is also iα-totally continuous.

Proof. Let -be any iα Vtotally continuous. Let -o iαbe any tw
and is an iα-totally continuous■ be an iα-totally continuous and-be an iα is an iα-totally continuous.

Proof.
Let be iα-totally continuous and Z Y g → : be iα-irresolute . Let V be iα-open set in Z. Since, g is an iα-irresolute, then g -1 (V) is an iα-open set in Y. Since, f is an iα-totally continuous, then f -1 (( g -1 (V))=(g o f ) -1 (V) is clopen set in X. Therefore, Z X gof → : is an iα-totally continuous■ is an iα-totally continuous and-is an iα is totally continuous.

Proof. Letcontinuous . Let -is an iα
totally continuous and -be iα Y X f → : V be an open set in Z. Since, g is an iα-continuous, then g -1 (V) is an iα-open set in Y. Since, f is an iα-totally continuous, then f -1 (g -1 (V))=(g o f ) -1 (V) is clopen set in X. Therefore, Z X gof → : is totally continuous■ is said to be iαcontra-continuous (resp. i-contra-continuous), if the inverse image of every open subset of Y is an iα-closed (resp. i-closed) set in X. X=Y={a,b,c},τ={Ø,{a} is an i-contracontinuous and iα-contra-continuous.

Proposition 4.12
Every contra-continuous mapping is an i-contra-continuous.

Proof.
Let be contra continuous mapping and V any open set in Y. Since, f is contra continuous, then f -1 (V) is closed sets in X. Since, every closed set is an i-closed set, then f -1 (V) is an i-closed set in X. Therefore, f is an i-contra-continuous■ Similarly we have the following results.

Proof. Clear since every semi-open set is an i-open set■
Proposition 4.14 Every contra α-continuous mapping is an i-contra-continuous.

Proof. Clear since every α-open set is an i-open set■
The converse of the propositions 4.12, 4.13 and 4.14 need not be true in general as shown in the following example Example 4.15 Let X=Y={a,b,c}, τ={Ø,{a,c},X}, iO(X)={Ø,{a},{c},{a,b},{a,c},{b, is is an i-contra continuous, but f is not contra-continuous, f is not contra semi-continuous, f is not contra α-continuous because for open subset f -1 {c}={c} is not closed in X, f -1 {c}={c}is not semi-closed in X and f -1 {c}={c}is not α-closed in X.

Proof.
Let be an i-contra continuous mapping and V any open set in Y. Since, f is an i-contra continuous, then f -1 (V) is an i-closed sets in X. Since, every iclosed set is an iα-closed, then f -1 (V) is an iα-closed set in X. Therefore, f is an iαcontra-continuous■ Remark 4. 17 The following example shows that iα-contra-continuous mapping need not be contra-continuous, contra semi-continuous, contra-α-continuous and i-contracontinuous mappings.

Theorem 4.19
Every totally continuous mapping is an iα-contra continuous.

Proof.
Let be totally continuous and V be any open set in Y. Since, f is totally continuous mapping, then f -1 (V) is clopen set in X, and hence closed, it follows iα-closed set. Therefore, f is an iα-contra-continuous■ The converse of the above theorem need not be true as shown in the following example

Separation Axioms with iα-open Set
In this section, we introduce some new weak of separation axioms with iα-open sets. (iv) iα-regular (resp. ultra regular [9]) if for each closed set F not containing a point in X can be separated by disjoint iα-open (resp. clopen) sets.
(v) clopen regular [10] if for each clopen set F not containing a point in X can be separated by disjoint open sets. (vi) iα-normal (resp. ultra normal [10], s-normal [13], α-normal [2]) if for each of nonempty disjoint closed sets in X can be separated by disjoint iα-open (resp. clopen, semiopen, α-open) sets. (vii) clopen normal [10] if for each of non-empty disjoint clopen sets in X can be separated by disjoint open sets. (viii) iα-T1/2 if every iα-closed is i-closed in X.

Theorem 5.4 if a mappingcontinuous mapping and the -contra-is an iα
space X is an iα-T1/2, then f is an i-contra-continuous.

Proof. Let . Yis any open set in Vcontinuous mapping and -racont-iα
Since, f is an iα-contra-continuous mapping, then f -1 (V) is an iα-closed in X. Since, X is an iα-T1/2, then f -1 (V) is i-closed in X. Therefore, f is an i-contra-continuous■ is an iα-totally continuous injection mapping and Y is an iα-T1, then X is clopen-T1.
Proof. Let x and y be any two distinct points in X. Since, f is an injective, we have f (x) and f (y)Y such that f (x)≠ f (y). Since, Y is an iα-T1, there exists iα-open sets U and V in Y such that f (x) U, f (y)  U and f (y) V, f (x)  V. Therefore, we have xf -1 (U),y f --1 (U) and yf --1 (V) and x f --1 (V), where f --1 (U) and f --1 (V) are clopen subsets of X because f is an iα-totally continuous. This shows that X is clopen-T1■ is an iα-totally continuous injection mapping and Y is an iα-To, then X is ultra-Hausdorff (UT2).
Proof. Let a and b be any pair of distinct points of X and f be an injective, then f (a)≠ f (b) in Y. Since Y is an iα-To, there exists iα-open set U containing f (a) but not f (b), then we have af --1 (U) and b f --1 (U). Since, f is an iα-totally continuous, then f --1 (U) is clopen in X. Also af --1 (U) and bX-f --1 (U). This implies every pair of distinct points of X can be separated by disjoint clopen sets in X. Therefore, X is ultra-Hausdorff■ be a closed iα-continuous injection mapping. If Y is an iαnormal, then X is an iα-normal.
Proof. Let F be a closed set not containing x. Since, f is closed, we have f (F) is a closed set in Y not containing f (x). Since, Y is an iα-regular, there exists disjoint iα-open sets A and B such that f (x)A and f (F)  B, which imply xf --1 (A) and F  f --1 (B), where f --1 (A) and f --1 (B) are clopen sets in X because f is totally continuous. Moreover, since f is an injective, we have f --1 (A) ∩ f --1 (B)= f --1 (A∩B)= f --1 (Ø)= Ø. Thus, for a pair of a point and a closed set not containing a point in X can be separated by disjoint clopen sets. Therefore, X is ultra-regular■ Theorem 5.10 Ifopen mapping from a -is totally continuous injective iα Y X f → : clopen regular space X into a space Y, then Y is an iα-regular.
Proof. Let F be a closed set in Y and y F. Take y =f (x). Since, f is totally continuous, and yf (V). Therefore, Y is an iα-regular■ is a totally continuous injective and iα-open mapping from clopen normal space X into a space Y, then Y is an iα-normal.