Functions almost contra-super-continuity in m-spaces

In this article, we study a generalizations of some class of functions that are in relation with the notions of continuity when we use the notions of minimal structures also its are characterized. Moreover we show that the notion of m-e∗-T1/2 spaces, given by Ekici [6], is a particular case of the m-(e∗)-T1/2 spaces when its are defined using the notion of m-generalized closed sets.


Introduction
The concepts of δ-open sets was introduced and studied by Velicko [27] in 1968, which is a stronger notion of open set.The notion of generalized closed (briefly g-closed) sets was introduced by Levine [13] in 1970.In 1987, P. Bhartacharyya et al. [2] introduced the notion of semi-generalized closed sets in Topology.Furthermore, the notion of quasi θ-continuous functions [11](resp.semi generalized continuous maps and semi-T 1/2 spaces [26], α-continuous and α-open mappings [16]) is introduced and studied.Later, in [21] and [22] Popa and Noiri introduced the notions of minimal structures.After this work, various mathematicians turned their attention in introducing and studying diverse classes of sets and functions defined on an structure, because this notions are a natural generalization of many well known results related with generalized sets and several weaker forms of Continuity.Each one of these classes of sets is, in turn, used in order to obtain different separation properties and new form of continuity (see [3], [4], [10], [11], [17], [20], [21] for details).E. Ekici [5] in 2004, studied the (δ-pre, s)-continuous functions on topological spaces and defined the m-e * -T 1/2 spaces if every m-e * -closed set is m-δ-closed.In this article we introduce and study the (m, m ′ )-almost contra-super-continuous, (m, m ′ )(a,s)-continuous, (m, m ′ )-(δ-semi,s)continuous, (m, m ′ )-2000 Mathematics Subject Classification: 54C10, 54D10 Luis Vásquez, Margot Salas Brown and Ennis Rosas (δ-pre,s)continuous, (m, m ′ )-(e,s)continuous, (m, m ′ )-(e * ,s)-continuous functions between spaces with minimal structure and study its relations with other class of functions and prove that the definition of m-e * -T 1/2 spaces given by Ekici is a particular case of the m-(e * )-T 1/2 spaces when it is defined in terms that each m-e * -generalized closed set is m-e * -closed.

Preliminaries
Let X be a nonempty set and let m ⊆ P (X), where P (X) denote the set of power of X.We say that m is an minimal structure on X (see [21] and [22]) if ∅ and X belong to m.The members of the minimal structure m are called m-open sets, and the pair (X, m) is called an m-space.The complement of an m-open set is called m-closed set.Definition 2.1 [14] Let (X, m) be an m-space and A ⊂ X, the m-interior of A and the m-closure of A are defined, respectively, as And satisfy the following properties: Proof: It follows from by Lemma 3.1 [21], [22]. 2 We can observe that, given a minimal structure m on a set X, if A ⊂ X, the m-int(A) is not necessarily an element of m, but we assume on m the condition that is closed under arbitrary unions (this condition is called the Maki condition), then immediately, we have that m-int(A) is an element of m, and hence Definition 2.3 Let (X, m) be an m-space.A subset A of X is said to be: Functions almost contra-super-continuity in m-spaces Observe that if the minimal structure is a topology, then the above concepts are the same as the concepts of regular open [25], semiopen [12], α-open [19], preopen [15] and β-open [1].Definition 2.4 [23]Let (X, m) be an m-space and A be a subset of Definition 2.5 [23]Let (X, m) be an m-space and A ⊂ X.A is said to be an Definition 2.6 Let (X, m) be an m-space and A ⊆ X.The m-r-kernel of A, denoted by m-r-ker(A), is defined as the intersection of all m-regular open sets that contain A, that is, Definition 2.7 Let (X, m) be an m-space and A ⊆ X.The m-δ-closure and the m-δ-interior of the set A, are defined, respectively, as: Let A be a subset of an m-space X.The following statements hold: Definition 2.9 A subset A of an m-space X is said to be: The complement of an m-δ-open set (respectively m-δ-semiopen set, m-δ-preopen set) is called m-δ-closed set (respectively m-δ-semiclosed set, m-δ-preclosed set).
Definition 2.10 A subset A of an m-space X is said to be: Example 2.13 Let R be the set of real number, x, y two distinct points in R and m = {R, ∅, {x}, {y}}.Let a be a point in R distinct of x and y.

The set
2. The set A = {x, a} is m-e-open but not is m-δ-preopen.

The set
Example 2.15 Let R be the set of real number, x, y two distinct points in R and m = {R, ∅, R \ {x}, R \ {y}}.

The set
The Proof: and we obtain that In 2007 Salas, M. et.al. [24] studied and generalized the separation axioms using minimal structure.Now we define the notions of m-T 1 spaces and m-T 2 spaces given in [23].
Definition 2.17 [23] Let (X, m) be an m-space, X is said to be: If (X, m) is an m space and consider the m spaces (X, m-e * O(X)), (X, m-eO(X)) and (X, m-aO(X).We obtain the concepts of m-e * -T 1 , m-e * -T 2 spaces (respectively m-e-T 1 , m-e-T 2 , m-a-T 1 , m-a-T 2 ) spaces, that are a natural generalizations of the definitions given by Ekici [9] when the minimal structure m is a topology.

Functions almost contra-super-continuous in m-spaces
Using the sets described in the above section, we define a new class of continuous functions between m spaces and we give some characterizations.Definition 3.1 Let (X, m), (Y, m ′ ) two m-spaces and f : X → Y be a function between m-spaces, f is said to be: If in the above definitions the minimal structures m and m ′ are topologies on X and Y respectively we obtain the classical concepts of function contra R-map [8], almost contra-super-continuous [6], (δ-semi, s)-continuous [7], (δ-pre, s)-continuous [5], (e * ,s)-continuous [9], (e, s)-continuous [9] and (a, s)-continuous [9] respectively.
The following theorem shows the existent relations between the different class of functions defined above.
Theorem 3.6 Let f : X → Y be a function between m-spaces.The following statements hold: Example 3.9 Let X = {a, b, c, d} and m = {∅, X, {a}, {c}, {a, b}, {a, c}, {a, b, c}, {a, c, d}}.Then the identity function  Proof: The concept of e * -T 1/2 spaces was introduced by Ekici [10] in the case of topological spaces, this concept characterize some classes of functions.Now we define a new class of spaces, the m-e * -T 1/2 spaces and characterize some class of functions.
Theorem 3.19 Let (X, m), (Y, m ′ ) m-spaces and f : X → Y be a function.If X is an m-e * -T 1/2 space, the following propositions are equivalent: Functions almost contra-super-continuity in m-spaces 25 ( Theorem 3.20 Let (Y, m ′ ) be an m ′ -regular space and f : X → Y be a function.
. By (7) x / ∈ m-e * -cl(f −1 (Y \ m ′ -cl(A))) and therefore there exists      In general topology, the notion of T 1/2 spaces is defined if every generalized closed set is closed.In 2007 Salas, M. Carpintero, C. and Rosas, E [24] studied and generalize these spaces using minimal structure.In 2007 Ekici [10] introduced the notion of T 1/2 spaces associated with the δ-closed sets.In this section, we compare the Definition of m − T 1/2 given by Salas, M. Carpintero, C. and Rosas, E [24] and the Definition 3.17.
Definition 5.1 [18] Let (X, m) be an m-space, A a subset of X, A is said to be an m-generalized closed set, abbreviate m-g-closed, if m-cl(A) ⊂ U whenever A ⊂ U and U ∈ m.Definition 5.2 [4] Let (X, m) be an m-space, X is said to be an m-T 1/2 space, if every m-g-closed set is m-closed.
The complement of an m-e-open set (respectively m-e * -open set, m-aopen set) is called m-e-closed set (respectively m-e * -closed set, m-aclosed set).The family of all m-e-open sets (respectively m-e * -open sets, m-a-open sets) are denoted by m-eO(X) (respectively m − e * O(X), m-aO(X)).The following diagram 1 shows the existence relation between the different sets defined above.(M)=Maki condition Example 2.11 Let R be the set of real number, x, y two distinct points in R and m = {R, ∅, {x}, {y}, R \ {x}, R \ {y}}.The set A = {x, y} is an m-δ-open but is not m-regular open neither m-open.Example 2.12 Let R be the set of real number, x, y two distinct points in R and m = {R, ∅, {x}, R \ {y}}.The set A = {x} is an m-open but is not m-δ-open.

Figure 1 :
Figure 1: Relation among the sets in an m-spaces

2 .
following theorem shows that the collection of all m-e * -open sets (respectively m-e-open sets, m-a-open sets) is an m structure that satisfy the Maki condition.Theorem 2.16 Let (X, m) be an m-space, the following statements hold: 1.The union of any collection of m-e * -open sets is an m-e * -open set.The union of any collection of m-e-open sets is an m-e-open set.3. The union of any collection of m-a-open sets is an m-a-open set.

α∈JU
α is an m-e * -open set.In analogue form follows (2) and (3). 2 We define the m-e-closure (respectively m-e * -closure, m-a-closure) of a subset A of X, denoted by m-e-cl(A) (respectively m-e * -cl(A), m-a-cl(A)), as the intersection of all m-e-closed sets (respectively m-e * -closed sets, m-a-closed sets) containing A. Now using the above theorem, we obtain in a natural form that the m-e-cl(A) (respectively m-e * -cl(A), m-a-cl(A)) is the smallest m-e-closed (respectively m-e *closed set, m-a-closed set) containing A.

1 .
m-T 1 if for each pair of different points x, y of X, there exist m-open sets M and N such that x ∈ M, y ∈ N and y / ∈ M and x / ∈ N .2. m-T 2 If for each pair of different points x, y of X there exist m-open sets M and N such that x ∈ M, y ∈ N and M ∩ N = ∅.

3 .
(m, m ′ )-almost e-continuous if f −1 (A) is an m-e-open set for all m ′ -regular open set A. 4. (m, m ′ )-almost a-continuous if f −1 (A) is an m-a-open set for all m ′ -regular open set A. Definition 3.14 An m-space (X, m) is said to be m-extremely disconnected if the m-closure of all m-open set of X is m-open.Example 3.15 Consider X = {a, b, c, d} and m = {∅, X, {a}, {b}, {c}, {d}, {b, c, d}, {a, c, d}, {a, b, d}, {a, b, c}}.The m-space (X, m) is extremely disconnected.If the range of a function is m-extremely disconnected, then the concepts of functions (m, m ′ )-(e * ,s)-continuous and (m, m ′ )-almost e * -continuous are equivalents.This fact we can see in the following theorem.Theorem 3.16 Let (X, m), (Y, m ′ ) two m-spaces and f : X → Y be a function.If (Y, m ′ ) is an m ′ -extremely disconnected, then f is (m, m ′ )-(e * ,s)-continuous if and only if f is (m, m ′ )-almost e * -continuous.

( 2 )
→ (11) Since any m-θ-semi open set is the union of m-regular closed and the result follows.

( 18 ) 2 Remark 3 . 22 2 Theorem 4 . 21 2 5.
→ (1) Let U ∈ m ′ -RO(Y ).Then U ∈ m ′ -P O(Y ) and therefore f −1 (U ) = f −1 (m ′ -int(m ′ -cl(U ))) is m-e * -closed set in X.In the same form as we characterize the functions (m, m ′ )-(e * ,s) -continuous, in the Theorem 3.21, we can obtain similar characterizations for functions (m, m ′ )-(e,s)-continuous (respectively (m, m ′ )-(a,s)-continuous) changing in the Theorem 3.21, e * by e (respectively a).Corollary 3.23 Let f : X → Y be a function between m-spaces.The following statements are equivalent: Theorem 4.20 Let f : X → Y be a function.If f is injective, (m, m ′ )-(e,s)continuous and Y is s-Urysohn, then X is an m-e-T 2 space.Proof: The proof is similar to the proof of the Theorem 4.19.Let f : X → Y be a function.If f is injective, (m, m ′ )-(a,s)continuous and Y is s-Urysohn, then X is m-a-T 2 space.Proof: The proof is similar to the proof of the Theorem 4.19.Generalized closed sets and T 1/2 spaces