Almost Continuous Functions in Topological Spaces

1 Introduction

semi ∗ - ℐ -open sets, pre ∗ - ℐ -open sets and e - ℐ -open sets in ideal topological spaces were studied by [1,2,3], respectively. In this paper, further properties and some relationships of these sets are investigated. Some relationships of pre ∗ - ℐ -open sets, semi ∗ - ℐ -open sets and e - ℐ -open sets in ideal topological spaces are discussed. Furthermore, decompositions of continuous functions have been introduced.

An ideal ℐ on a nonempty set X is a nonempty collection of subsets of X , which satisfies the following conditions: Q ∈ ℐ and H ⊂ Q imply H ∈ ℐ ; Q ∈ ℐ and H ∈ ℐ imply Q ∪ H ∈ ℐ [4]. Applications to various fields were further investigated by Jankovic and Hamlett [5]; Mukherjee et al. [6]; Arenas et al. [7]; Nasef and Mahmoud [8]; Modak and Noiri [9,10,11]; Zahran and Ghareeb [12], etc. Given a topological space ( X , τ ) with an ideal ℐ on X and if ℘ ( X ) is the set of all subsets of X , a set operator ( . ) ∗ : ℘ ( X ) → ℘ ( X ) , called the local function [4] of Q with respect to τ and ℐ is defined as follows: for Q ⊆ X ,

Q ∗ ( ℐ , τ ) = { x ∈ X ∣ V ∩ Q ∉ ℐ for every V ∈ τ ( x ) } ,

where τ ( x ) = { V ∈ τ ∣ x ∈ V } . Furthermore, C l ∗ ( Q ) = Q ∪ Q ∗ ( ℐ , τ ) defines a Kuratowski closure operator for the topology τ ∗ , called the ∗ -topology, finer than τ . When there is no chance for confusion, we will simply write Q ∗ for Q ∗ ( ℐ , τ ) . X ∗ is often a proper subset of X . By a space, we always mean a topological space ( X , τ ) with no separation properties assumed. If Q ⊂ X , C l ( Q ) and I n t ( Q ) will denote the closure and interior of Q in ( X , τ ) , respectively.

A subset A of a space ( X , τ ) is said to be regular open (resp. regular closed) [13] if A = I n t ( C l ( A ) ) (resp. A = C l ( I n t ( A ) ) ). A set S is called δ -open [14] if for each x ∈ S , there exists a regular open set K such that x ∈ K ⊂ S . The union of all δ -open subsets of S is called δ -interior [14] of S in X and is denoted by δ i n t ( S ) , S is δ -open if I n t δ ( A ) = A . The complement of a δ -open set is called δ -closed [14]. δ c l ( S ) = { x ∣ U ∈ U ( x ) ⇒ i n t ( c l ( U ) ) ∩ S ≠ ∅ } , and a set S is δ -closed if and only if S = δ c l ( S ) .

A subset Q of an ideal topological space ( X , τ , ℐ ) is said to be R - ℐ -open (resp. R - ℐ -closed) [15] if Q = I n t ( C l ∗ ( Q ) ) (resp. Q = C l ( I n t ∗ ( Q ) ) . A point x ∈ X is called a δ - ℐ -cluster point of Q if I n t ( C l ∗ ( V ) ) ∩ Q ≠ ∅ for each open set V containing x . The family of all δ - ℐ -cluster points of Q is called the δ - ℐ -closure of Q and is denoted by δ C l ℐ ( Q ) . The set δ - ℐ -interior of Q is the union of all R - ℐ -open sets of X contained in Q and it is denoted by δ I n t ℐ ( Q ) . Q is said to be δ - ℐ -closed (resp. δ - ℐ -open) if δ C l ℐ ( Q ) = Q (resp. δ I n t ℐ ( Q ) = Q ) [15]. δ - ℐ -open sets form a topology τ δ - ℐ and that it is coarser than τ .

A subset Q of an ideal topological space ( X , τ , ℐ ) is said to be e - ℐ -open [3,16] if Q ⊂ C l ( δ I n t ℐ ( Q ) ) ∪ I n t ( δ C l ℐ ( Q ) ) . The complement of an e - ℐ -open set is called an e - ℐ -closed set [17,18]. The family of all e - ℐ -open sets containing x of X will be denoted by E O ℐ ( X , x ) . All of the above concepts may be defined with the help of grill, and some of them have been defined by Modak in [10].

The aim of this paper is to introduce one class of functions, namely, almost e - ℐ -continuous functions containing the class of almost e -continuous functions by utilizing the notions of e - ℐ -open sets due to Al-Omeri et al. [3]. We investigate several properties of this class. The class of almost e - ℐ -continuity is a generalization of almost pre ∗ - ℐ -continuity and almost semi ∗ - ℐ -continuity. At the same time, the class of almost e - ℐ -continuity is a generalization of almost e -continuity.

This paper consists of eight sections. In Section 3, we introduce some new types of continuity such as e - ℐ -continuous functions. In Section 4, we investigate not only some of their properties but also relationships between almost e - ℐ -continuity and separation axioms. In Section 5, we introduce and study various types of continuity and connectedness and also almost e - ℐ -continuity. In Section 6, we introduce strongly almost e - ℐ -closed graphs and relationships between almost e - ℐ -continuity and strongly almost e - ℐ -closed graphs are obtained. Moreover, several characterizations and properties of strongly almost e - ℐ -closed graphs are investigated. In Section 7, we investigate relationships between almost e - ℐ -continuity and e - ℐ -compactness. In Section 8, we deal with relationships among several functions, almost e - ℐ -continuous, almost- ℐ -continuous, almost δ α - ℐ -continuous, almost semi ∗ - ℐ -continuous, almost pre ∗ - ℐ -continuous, almost δ β ℐ -continuous, almost- A ℐ R -continuous and continuous functions.

2 Preliminaries

Throughout the present paper, spaces always mean topological spaces and f : ( X , τ , ℐ ) ⟶ ( Y , σ ) denotes a function f of an ideal topological space ( X , τ , ℐ ) into a space ( Y , σ ) .

Definition 2.1

A subset U of an ideal topological space ( X , τ , ℐ ) is said to be

1. δ α - ℐ -open [1] if U ⊂ I n t ( C l ( δ I n t ℐ ( U ) ) ) .

2. semi ∗ - ℐ -open [1] if U ⊂ C l ( δ I n t ℐ ( U ) ) .

3. pre ∗ - ℐ -open [2] if U ⊆ I n t ( δ C l ℐ ( U ) ) .

4. e - ℐ -open [3] if U ⊂ C l ( δ I n t ℐ ( U ) ) ∪ I n t ( δ C l ℐ ( U ) ) .

5. δ β ℐ -open if U ⊆ C l ( I n t ( δ C l ℐ ( U ) ) ) .

6. A ℐ R -set if U = O ∩ F , where O is open and F is R - ℐ -closed.

7. a -open [20] if A ⊂ I n t ( C l ( δ I n t ( A ) ) ) .

Remark 2.2

Al-Omeri et al. [3] obtained Diagram I by using the notions in Definition 2.1. Furthermore, they showed via examples that the converses of implications are not true in general.

The e - ℐ -interior (resp. semi ∗ - ℐ -interior [1], pre ∗ - ℐ -interior [2]) of U , denoted by I n t e ∗ ( U ) (resp. S ∗ ℐ I n t ( U ) , P ∗ ℐ I n t ( U ) ), is defined by the union of all e - ℐ -open [3] (resp. semi ∗ - ℐ -open [1], pre ∗ - ℐ -open [2]) sets contained in U . For a subset U of an ideal topological space ( X , τ , ℐ ) , the intersection of all e - ℐ -closed (resp. semi ∗ - ℐ -closed [1], pre ∗ - ℐ -closed [2]) sets containing U is called the e - ℐ -closure (resp. semi ∗ - ℐ -closure [1], pre ∗ - ℐ -closure [2]) of U and is denoted by C l e ∗ ( U ) (resp. S ∗ ℐ C l ( U ) , P ∗ ℐ C l ( U ) ).

The family of all R - ℐ -open (resp. δ α - ℐ -open, pre ∗ - ℐ -open, semi ∗ - ℐ -open, e - ℐ -open, δ β ℐ -open) sets and A ℐ R -sets of X will be denoted by R O ℐ ( X ) (resp. δ α O ℐ ( X ) , P O ℐ ( X ) , S O ℐ ( X ) , E O ℐ ( X ) , β O ℐ ( X ) ) and A O ℐ ( X ) . Similarly, the family of all R - ℐ -closed (resp. e - ℐ -closed) sets in X will be denoted by R C ℐ ( X ) (resp. E C ℐ ( X ) ).

Definition 2.3

A function f : ( X , τ , ℐ ) ⟶ ( Y , σ , J ) is said to be almost- ℐ -continuous [15] if for each x ∈ X and each open set V of Y containing f ( x ) , there exists an open set U containing x such that f ( U ) ⊂ I n t ( C l ∗ ( V ) ) .

For the function which is mentioned above, we will use the abbreviation: a. ℐ .c.

3 Almost e - ℐ -continuous functions

We obtain some characterizations of almost e - ℐ -continuous functions and give some properties of these functions.

Definition 3.1

A function f : ( X , τ , ℐ ) ⟶ ( Y , σ , J ) is said to be almost δ α - ℐ -continuous (resp. almost pre ∗ - ℐ -continuous, almost semi ∗ - ℐ -continuous, almost δ β ℐ -continuous, almost A ℐ R -continuous) if f − 1 ( Q ) ∈ δ α O ℐ ( X ) (resp. f − 1 ( Q ) ∈ P O ℐ ( X ) , f − 1 ( Q ) ∈ S O ℐ ( X ) , f − 1 ( Q ) ∈ β O ℐ ( X ) , f − 1 ( Q ) ∈ A O ℐ ( X ) ) for every Q ∈ R O J ( Y ) . For functions which are mentioned above, we will use the abbreviations: a . δ α . c , a . p ℐ . c ., a . s ℐ . c , a . δ β . c , a . A ℐ R . c ., respectively.

We have Diagram II among the functions defined above.

The converses of implications are not true in general.

Definition 3.2

A function f : ( X , τ , ℐ ) ⟶ ( Y , σ , J ) is said to be almost e - ℐ -continuous (briefly a.e.c.) for each x ∈ X if for each Q ⊆ R O J ( Y ) containing f ( x ) , there exists H ∈ E O ℐ ( X ) containing x such that f ( H ) ⊆ Q .

Theorem 3.3

For a function f : ( X , τ , ℐ ) ⟶ ( Y , σ , J ) , the following properties are equivalent:

1. f is a.e.c.;

2. For each x ∈ X and each Q containing f ( x ) , there exists U ∈ E O ℐ ( X ) containing x such that f ( U ) ⊆ I n t ( δ C l J ( Q ) ) ;

3. f − 1 ( V ) ∈ E C ℐ ( X ) for every V ∈ R C J ( Y ) ;

4. f − 1 ( Q ) ∈ E O ℐ ( X ) for every Q ∈ R O J ( Y ) .

Proof

The proof is obvious and so is thus omitted.□

Theorem 3.4

For a function f : ( X , τ , ℐ ) ⟶ ( Y , σ , J ) , the following properties are equivalent:

1. f is a.e.c.;

2. f ( C l e ∗ ( Q ) ) ⊆ δ C l J ( f ( Q ) ) for every subset Q of X ;

3. C l e ∗ ( f − 1 ( H ) ) ⊆ f − 1 ( δ C l J ( H ) ) for every subset H of Y ;

4. f − 1 ( T ) ∈ E C ℐ ( X ) for every δ - J -closed set subset T of Y ;

5. f − 1 ( Q ) ∈ E O ℐ ( X ) for every δ - J -open set Q of Y .

Proof

( 1 ) ⇒ ( 2 ) : Let Q be a subset of X . Since δ C l J ( f ( Q ) ) is a δ - J -closed set in Y , it is denoted by ∩ { T α : f ( Q ) ⊂ T α , T α ∈ R C J ( Y ) , α ∈ Δ } , where Δ is an index set. By Theorem 3.3, and by Lemma 2.2.(3)+(5) in [15], we have Q ⊆ f − 1 ( δ C l J ( f ( Q ) ) ) = ∩ { f − 1 ( T α ) : α ∈ Δ } and hence C l e ∗ ( Q ) ⊆ f − 1 ( δ C l J ( f ( Q ) ) ) . Hence, we obtain f ( C l e ∗ ( Q ) ) ⊆ δ C l J ( f ( Q ) ) .

( 2 ) ⇒ ( 3 ) : Let H be a subset of Y . We have f ( C l e ∗ ( f − 1 ( H ) ) ) ⊆ δ C l J ( f ( f − 1 ( H ) ) ) ⊆ δ C l J ( H ) and hence C l e ∗ ( f − 1 ( H ) ) ⊆ f − 1 ( δ C l J ( H ) ) .

( 3 ) ⇒ ( 4 ) : Let T be any δ - J -closed set of Y . We have C l e ∗ ( f − 1 ( T ) ) ⊆ f − 1 ( δ C l J ( T ) ) = f − 1 ( T ) and f − 1 ( T ) is e - ℐ -closed in X .

( 4 ) ⇒ ( 5 ) : Let Q be any δ - J -open set of Y . According to (4), we have f − 1 ( Y − Q ) = X − f − 1 ( Q ) ∈ E C ℐ ( X ) and so f − 1 ( Q ) ∈ E O ℐ ( X ) .

( 5 ) ⇒ ( 1 ) : Let Q be any R - ℐ -open set of Y . Since Q is δ - J -open in Y , f − 1 ( Q ) ∈ E O ℐ ( X ) and hence by Theorem 3.3, f is a.e.c.□

Definition 3.5

[15] A function f : ( X , τ , ℐ ) ⟶ ( Y , σ , J ) is said to be δ ℐ -continuous if for each x ∈ X and each open neighborhood Q of f ( x ) , there exists an open neighborhood H of x such that f ( I n t ( C l ∗ ( H ) ) ) ⊆ I n t ( C l ∗ ( Q ) ) .

Theorem 3.6

[15] A function f : ( X , τ , ℐ ) ⟶ ( Y , σ , J ) is δ ℐ -continuous if and only if for every R - ℐ -open set Q of Y , f − 1 ( Q ) is δ - ℐ -open in X .

Definition 3.7

[19] A function f : ( X , τ , ℐ ) ⟶ ( Y , σ , J ) is said to be e - ℐ -irresolute if f − 1 ( H ) ∈ E O ℐ ( X ) for each H ∈ E O J ( Y ) .

Definition 3.8

A function f : ( X , τ , ℐ ) ⟶ ( Y , σ , J ) is said to be an R - ℐ -map if f − 1 ( Q ) ∈ R O ℐ ( X ) for every H ∈ R O J ( Y ) .

Theorem 3.9

Let f : ( X , τ , ℐ ) ⟶ ( Y , σ 1 , J ) and g : ( Y , σ 1 , J ) ⟶ ( Z , σ 2 , K ) be functions. For the composition g ∘ f : ( X , τ , ℐ ) ⟶ ( Z , σ 2 , K ) , the following properties hold:

1. If f is a.e.c. and g is an R - ℐ -map, then g ∘ f is a.e.c.;

2. If f is e - ℐ -irresolute and g is a.e.c., then g ∘ f is a.e.c.;

3. If f is a.e.c. and g is a . ℐ . c , then g ∘ f is a.e.c.;

4. If f is a.e.c. and g is δ ℐ -continuous, then g ∘ f is a.e.c.

Proof

The proof of (4) is obtained by using Theorems 3.6, 3.3 and 3.4. The proofs of (1)–(3) are obtained immediately as consequences of the definitions and it is thus omitted.□

Let ( X , τ , ℐ ) be an ideal topological space and B be a subset of X . If we denote by ℐ / B the relative ideal on B , then ℐ / B = { B ∩ I : I ∈ ℐ } .

The following lemma is due to [3].

Lemma 3.10

Let ( X , τ , ℐ ) be an ideal topological space and let Q , U ⊆ X . If Q is an e - ℐ -open set and U ∈ τ , then Q ∩ U ∈ E O ℐ ( U , τ / U , ℐ / U ) .

Theorem 3.11

If f : ( X , τ , ℐ ) ⟶ ( Y , σ , J ) is a.e.c. and U ∈ τ , then the restriction f ∣ U : ( U , τ / U , ℐ / U ) ⟶ ( Y , σ , J ) is a.e.c.

Proof

Let Q be any R - J -open set of Y . By Theorem 3.3, we have f − 1 ( Q ) ∈ E O ℐ ( X ) and hence ( f ∣ U ) − 1 ( Q ) = f − 1 ( Q ) ∩ U ∈ E O ℐ ( U ) by Lemma 3.10. Thus, it follows from Theorem 3.3 that f ∣ U is a.e.c.□

4 Separation axioms

In this section, we obtain some relationships and several properties between separation axioms and almost e - ℐ -continuous functions.

Definition 4.1

[19] An ideal topological space ( X , τ , ℐ ) is said to be e - ℐ - T 1 if for each pair of distinct points x and y in X , there exist e - ℐ -open sets Q and H containing x and y , respectively, such that y ∉ Q and x ∉ H .

Definition 4.2

[19] An ideal topological space ( X , τ , ℐ ) is said to be e - ℐ - T 2 if for each pair of distinct points x and y in X , there exist e - ℐ -open sets Q and H containing x and y , respectively, such that Q ∩ H = ∅ .

Definition 4.3

An ideal topological space ( X , τ , ℐ ) is said to be R - ℐ - T 1 if for each pair of distinct points x and y in X , there exist R - ℐ -open sets Q and H containing x and y , respectively, such that y ∉ Q and x ∉ H .

Theorem 4.4

If f : ( X , τ , ℐ ) ⟶ ( Y , σ , J ) is an a.e.c. injection and Y is R - J - T 1 , then X is e - ℐ - T 1 .

Proof

Assume that Y is R - J - T 1 . For any distinct points x and y in X , there exist Q , H ∈ R O J ( Y ) such that f ( x ) ∈ Q , f ( y ) ∉ Q , f ( x ) ∉ H and f ( y ) ∈ H . Since f is a.e.c., by Definition 3.2, there exist K , L ∈ E O ℐ ( X ) such that x ∈ K , y ∈ L , f ( K ) ⊆ Q and f ( L ) ⊆ H . Thus, we obtain y ∉ K , x ∉ L . This shows that X is e - ℐ - T 1 .□

Theorem 4.5

If f : ( X , τ , ℐ ) ⟶ ( Y , σ , J ) is an a.e.c. injection and Y is T 2 , then X is e - ℐ - T 2 .

Proof

For any pair of distinct points x and y in X , there exist disjoint open sets Q , H ⊆ Y such that f ( x ) ∈ Q , f ( y ) ∈ H . Since f is a.e.c., there exist e - ℐ -open sets K , L ⊆ X containing x and y , respectively, such that f ( K ) ⊆ I n t ( δ C l J ( Q ) ) and f ( L ) ⊆ I n t ( δ C l J ( H ) ) . Since Q and H are disjoint, we have I n t ( δ C l J ( H ) ) ∩ I n t ( δ C l J ( H ) ) = ∅ , hence K ∩ L = ∅ . This shows that X is e - ℐ - T 2 .□

Definition 4.6

Let ( X , τ , ℐ ) be an ideal topological space.

1. τ e = { Q ⊆ X : Q ∩ H ∈ E O ℐ ( X ) whenever H ∈ E O ℐ ( X ) } .

2. ( X , τ , ℐ ) is called an E ∗ -space if E O ℐ ( X ) is closed under finite intersection.

3. ( τ e ) s is the topology having E O ℐ ( X ) as the subbase.

Lemma 4.7

If ( X , τ , ℐ ) is an ideal topological space, then the following properties hold:

1. τ e is a topology for X ;

2. τ ⊂ E O ℐ ( X ) ⊂ τ e ⊂ ( τ e ) s ;

3. If ( X , τ , ℐ ) is an E ∗ -space, then τ e = E O ℐ ( X ) = ( τ e ) s ;

4. ( τ e ) s is the smallest topology containing E O ℐ ( X ) .

Proof

1. 1. It is obvious that ∅ , X ∈ τ e .

   2. Let Q 1 , Q 2 ∈ τ e . Then Q 1 ∩ H , Q 2 ∩ H ∈ E O ℐ ( X ) whenever H ∈ E O ℐ ( X ) . Hence, ( Q 1 ∩ Q 2 ) ∩ H = Q 1 ∩ ( Q 2 ∩ H ) ∈ E O ℐ ( X ) whenever H ∈ E O ℐ ( X ) . Therefore, Q 1 ∩ Q 2 ∈ τ e .

   3. Let Q α ∈ τ e for every α ∈ Δ . Then Q α ∩ H ∈ E O ℐ ( X ) whenever H ∈ E O ℐ ( X ) for every α ∈ Δ . Since ( ∪ { Q α : α ∈ Δ } ) ∩ H = ∪ { ( Q α ∩ H ) : α ∈ Δ } and E O ℐ ( X ) is closed under arbitrary union, ( ∪ Q α ) ∩ H ∈ E O ℐ ( X ) and ∪ Q α ∈ τ e .

2. For every U ∈ τ , U ∩ H ∈ E O ℐ ( X ) whenever H ∈ E O ℐ ( X ) by Proposition 2.7 of [3]. Since X ∈ E O ℐ ( X ) , for every Q ∈ τ e , Q = Q ∩ X ∈ E O ℐ ( X ) .

3. The proofs are obvious.□

Theorem 4.8

Let ( X , τ , ℐ ) be an E ∗ -space. If f , g : ( X , τ , ℐ ) ⟶ ( Y , σ , J ) are a.e.c. and ( Y , σ ) is Hausdorff, then Δ = { x ∈ X ∣ f ( x ) = g ( x ) } is e - ℐ -closed in ( X , τ , ℐ ) .

Proof

Since f , g : ( X , τ , ℐ ) ⟶ ( Y , σ , J ) are a.e.c., by Theorems 3.4 and 8.7 f , g : ( X , τ e , ℐ ) ⟶ ( Y , σ s , J ) are continuous. Since ( Y , σ δ J , J ) is Hausdorff, Δ = { x ∈ X ∣ f ( x ) = g ( x ) } is closed in ( X , τ e , ℐ ) and hence e - ℐ -closed in ( X , τ , ℐ ) .□

Recall that a function f : ( X , τ ) ⟶ ( Y , σ s , J ) is called weakly continuous [21] if for each x ∈ X and each V ∈ σ containing f ( x ) , there exists U ∈ τ containing x such that f ( U ) ⊂ C l ( V ) . For weakly continuous functions, we will use the abbreviation w.c.

Theorem 4.9

Let ( X , τ , ℐ ) be an E ∗ -space and let f , g : ( X , τ , ℐ ) ⟶ ( Y , σ , J ) be functions and ( Y , σ , J ) Hausdorff. If f is w.c. and g is a.e.c., then Δ = { x ∈ X ∣ f ( x ) = g ( x ) } is e - ℐ -closed in X .

Proof

Assume that x ∉ Δ . Then f ( x ) ≠ g ( x ) . Since Y is Hausdorff, there exist open sets Q and H of Y such that f ( x ) ∈ Q , g ( x ) ∈ H and Q ∩ H = ∅ and hence C l ( Q ) ∩ I n t ( δ C l J ( H ) ) = ∅ . Since f is w.c., there exists an open set K containing x such that f ( K ) ⊆ C l ( Q ) . Since g is a.e.c., there exists L ∈ E O ℐ ( X ) containing x such that g ( L ) ⊆ I n t ( δ C l ℐ ( H ) ) . Put P = K ∩ L , then P ∈ E O ℐ ( X ) by Proposition 2.7 of [3] and P ∩ Δ = ∅ . Therefore, we obtain x ∉ C l e ∗ ( Δ ) . This shows that Δ is e - ℐ -closed in X .□

5 Connectedness

In this section, we study the relationships between almost e - ℐ -continuous functions and e - ℐ -connectedness.

Definition 5.1

[19] An ideal topological space ( X , τ , ℐ ) is said to be e - ℐ -connected if X is not the union of two disjoint non empty e - ℐ -open subsets of X .

This is a stronger form of connectedness and various types of stronger and weaker forms of connectedness have been discussed in [22,23,24, 25,26]

Theorem 5.2

If f : ( X , τ , ℐ ) ⟶ ( Y , σ , J ) is an a.e.c. surjection and X is e - ℐ -connected, then Y is connected.

Proof

Suppose that Y is not a connected space. Then Y can be written as Y = Q 0 ∪ H 0 such that Q 0 and H 0 are disjoint non empty open sets. Let Q = I n t ( C l ∗ ( Q 0 ) ) and H = I n t ( C l ∗ ( H 0 ) ) . Then Q and H are disjoint nonempty R - ℐ -open sets such that Y = Q ∪ H . Since f is a.e.c., then f − 1 ( Q ) and f − 1 ( H ) are e - ℐ -open sets of X . We have X = f − 1 ( Q ) ∪ f − 1 ( H ) such that f − 1 ( Q ) and f − 1 ( H ) are non empty disjoint since f is surjective. This shows that X is not e - ℐ -connected. Hence, Y is connected.□

6 Strongly almost e - ℐ -closed graphs

Definition 6.1

[19] The graph G ( f ) of a function f : ( X , τ , ℐ ) ⟶ ( Y , σ ) is said to be strongly almost e - ℐ -closed if for each ( x , y ) ∈ X × Y − G ( f ) , there exist Q ∈ E O ℐ ( X , x ) and a regular open set H in Y containing y such that ( Q × H ) ∩ G ( f ) = ∅ .

Lemma 6.2

A function f : ( X , τ , ℐ ) ⟶ ( Y , σ ) has the strongly almost e - ℐ -closed graph if and only if for each x ∈ X and y ∈ Y such that f ( x ) ≠ y , there exist Q ∈ E O ℐ ( X , x ) and a regular open set H containing y such that f ( Q ) ∩ H = ∅ .

Proof

The proof is obtained immediately from Definition 6.1.□

Theorem 6.3

If f : ( X , τ , ℐ ) ⟶ ( Y , σ ) is w.c. and Y is Hausdorff, then G ( f ) is strongly almost e - ℐ -closed.

Proof

Suppose that ( x , y ) is any point of ( X × Y ) − G ( f ) . Then y ≠ f ( x ) . But Y is Hausdorff and hence there exist open sets B 1 and B 2 in Y such that y ∈ B 1 , f ( x ) ∈ B 2 and B 1 ∩ B 2 = ∅ . Since B 1 and B 2 are disjoint and open, we obtain I n t ( C l ( B 1 ) ) ∩ C l ( B 2 ) = ∅ . Since f is w.c., then there exists Q ∈ τ containing x such that f ( Q ) ⊆ C l ( B 2 ) . Hence, f ( Q ) ∩ I n t ( C l ( B 1 ) ) = ∅ . It follows from Lemma 6.2 that G ( f ) is strongly almost e - ℐ -closed.□

Theorem 6.4

If f : ( X , τ , ℐ ) ⟶ ( Y , σ ) has a strongly almost e - ℐ -closed graph G ( f ) and f is injective, then X is e - ℐ - T 1 .

Proof

Let x and y be any two distinct points of X . Then we have ( x , f ( y ) ) ∈ ( X × Y ) − G ( f ) . Since G ( f ) is strongly almost e - ℐ -closed, there exist Q ∈ E O ℐ ( X , x ) and a regular open set H containing f ( y ) such that f ( Q ) ∩ H = ∅ by using Lemma 6.2. Therefore, f − 1 ( H ) ∩ Q = ∅ and hence we have y ∉ Q . This shows that X is e - ℐ - T 1 .□

Definition 6.5

[19] A function f : ( X , τ , ℐ ) ⟶ ( Y , σ , J ) is said to be pre e - ℐ -open if the image of each e - ℐ -open set of X is an e - J -open set in Y .

Theorem 6.6

Let f : ( X , τ , ℐ ) ⟶ ( Y , σ , J ) have a strongly almost e - ℐ -closed graph G ( f ) . If f is a surjective pre e - ℐ -open function, then Y is e - J - T 2 .

Proof

Let y 1 and y 2 be any distinct points of Y . Since f is surjection, f ( x ) = y 1 for some x ∈ X and ( x , y 2 ) ∈ ( X × Y ) − G ( f ) . Since G ( f ) is strongly almost e - ℐ -closed, by Lemma 6.2, there exist Q ∈ E O ℐ ( X , x ) and a regular open set H containing y 2 such that f ( Q ) ∩ H = ∅ . Since f is pre e - ℐ -open, then f ( Q ) is an e - J -open in Y such that f ( x ) = y 1 ∈ f ( Q ) . This implies that Y is e - J - T 2 .□

7 Compactness

Recall that a subset Q of a space X is said to be e - ℐ -compact [3] relative to X (resp. N -closed relative to X [27]) if every cover of Q by e - ℐ -open (resp. regular open) sets of X has a finite subcover.

Theorem 7.1

If f : ( X , τ , ℐ ) ⟶ ( Y , σ , J ) is a.e.c. and Q is e - ℐ -compact relative to X , then f ( Q ) is N -closed relative to Y .

Proof

Let { F α : α ∈ Δ } be any cover of f ( Q ) by regular open sets of Y . Since f is a.e.c., by Theorem 3.3 { f − 1 ( F α ) : α ∈ Δ } is a cover of Q by e - ℐ -open sets of X . Since Q is e - ℐ -compact relative to X , there exists a finite subset Δ 0 of Δ such that Q ⊆ ∪ { f − 1 ( F α ) : α ∈ Δ 0 } . Therefore, we obtain f ( Q ) ⊆ ∪ { F α : α ∈ Δ 0 } . This shows that f ( Q ) is N -closed relative to Y .□

Corollary 7.2

If f : ( X , τ , ℐ ) ⟶ ( Y , σ , J ) is an a.e.c. surjection and X is e - ℐ -compact, then Y is nearly compact.

Theorem 7.3

If f : ( X , τ , ℐ ) ⟶ ( Y , σ , J ) has a strongly almost e - ℐ -closed graph, then f ( Q ) is δ - ℐ -closed in Y for each subset Q which is e - ℐ -compact relative to X .

Proof

Assume that y ∉ f ( Q ) . Then ( x , y ) ∈ ( X × Y ) − G ( f ) for each x ∈ Q . Since G ( f ) is the strongly almost e - ℐ -closed graph, there exist V x ∈ E O ℐ ( X , x ) containing x and a regular open set U x of Y containing y such that f ( V x ) ∩ U x = ∅ . The family { V x : x ∈ Q } is a cover of Q by e - ℐ -open sets of X . Since Q is e - ℐ -compact relative to X , there exists a finite subset Q 0 of Q such that Q ⊆ ∪ { V x : x ∈ Q 0 } . Set U = ∩ { U x : x ∈ Q 0 } . Then U is a regular-open set in Y containing y . Therefore, we have f ( Q ) ∩ U ⊆ ∪ { f ( V x ) : x ∈ Q 0 } ∩ U ⊆ ∪ { f ( V x ) ∩ U : x ∈ Q 0 } = ∅ . It follows that y ∉ δ C l J f ( Q ) . Therefore, f ( Q ) is δ - ℐ -closed in Y .□

8 Comparisons and examples

In this section, we investigate relationships between almost e - ℐ -continuity and other related functions.

Proposition 8.1

Let f : ( X , τ , ℐ ) ⟶ ( Y , σ , J ) be a function. Then, the following properties hold:

1. If f is a . p ℐ . c ., then it is a.e.c.;

2. If f is a . s ℐ . c , then it is a.e.c.;

3. If f is a.e.c., then it is a . δ β . c .

Proof

It follows immediately from Theorem 3.3, Remark 2.2 and Definition 3.1.□

The converses of each statement in Proposition 8.1 need not be true as shown in the following examples, respectively.

Example 8.2

X = Y = { a , b , c , d } , σ = { ∅ , X , { a } , { b } , { c } , { a , b } , { a , c } , { b , c } , { a , b , c } } , τ = { ∅ , X , { a } , { b } , { a , b } } and ℐ = J = { Ø , { a } } .

1. Let f : ( X , τ , ℐ ) ⟶ ( Y , σ , J ) be a function defined as follows: f ( a ) = f ( b ) = f ( c ) = a and f ( d ) = d . Then, f is a . p ℐ . c and hence a.e.c. However, f is not a . s ℐ . c since there exists { a } ∈ R O ℐ ( X ) such that f − 1 ( { a } ) = { a , b , c } is not semi ∗ - ℐ -open

2. Let f : ( X , τ , ℐ ) ⟶ ( Y , σ , J ) be a function defined as follows: f ( a ) = f ( b ) = a and f ( c ) = f ( d ) = c . Then, f is a . δ β . c but is not a.e.c. since there exists { a } ∈ R O ℐ ( X ) such that f − 1 ( { a } ) = { a , b } is not e - ℐ -open in X .

Example 8.3

Let X = Y = { a , b , c , d } , σ = { ∅ , X , { a } , { b } , { c } , { a , b } , { a , c } , { b , c } , { a , b , c } } , τ = { ∅ , X , { a } , { b } , { a , b } } and ℐ = J = { Ø , { b } } . Let f : ( X , τ , ℐ ) ⟶ ( Y , σ , J ) be a function defined as follows f ( a ) = f ( c ) = b , f ( b ) = a and f ( d ) = d . Then, f is a . s ℐ . c and hence a.e.c. However, f is not a . p ℐ . c since there exists { b } ∈ R O ℐ ( X ) such that f − 1 ( { b } ) = { a , c } is not P r e ∗ - ℐ -open in X .

We recall that a space ( X , σ , ℐ ) is said to be submaximal [28] if every dense subset of X is open in X . A space ( X , τ , ℐ ) is said to be extremally disconnected [29] if the closure of every open set of X is open in X .

Definition 8.4

[30] A function f : ( X , τ , ℐ ) ⟶ ( Y , σ , J ) is said to be e -continuous if for each open set Q of ( Y , σ ) , f − 1 ( Q ) is e -open.

Theorem 8.5

Let f : ( X , τ , ℐ ) ⟶ ( Y , σ , J ) be a function such that X is submaximal and extremally disconnected. Then, the following properties are equivalent:

1. f is a . ℐ . c ;

2. f is a . δ α . c ;

3. f is a . s ℐ . c ;

4. f is a . p ℐ . c ;

5. f is a.e.c.;

6. f is a . δ β . c .

Proof

It is shown in Corollary 5.26 of [3] that if X is submaximal and extremally disconnected, then P O ℐ ( X , τ ) = S O ℐ ( X , τ ) = δ α O ℐ ( X , τ ) = τ . The proof is obvious by the fact.□

The converses of the following implications are not true in general.

Lemma 8.6

[31] If ( X , τ ) is resolvable, then τ e is discrete.

Theorem 8.7

Let ( X , τ , ℐ ) be an E ∗ -space. For a function f : ( X , τ , ℐ ) ⟶ ( Y , σ , J ) , the following properties are equivalent:

1. f : ( X , τ , ℐ ) ⟶ ( Y , σ , J ) is a.e.c.;

2. f : ( X , τ , ℐ ) ⟶ ( Y , σ s ) is e - ℐ -continuous;

3. f : ( X , τ e ) ⟶ ( Y , σ , J ) is a . ℐ . c ;

4. f : ( X , τ e ) ⟶ ( Y , σ s ) is continuous.

Proof

Every Q ∈ σ s is the union of regular open sets of ( Y , σ ) . Therefore, it is obvious that (1) and (3) are equivalent to (4) and also (1) and (2) are equivalent.□

Recall that a space X is said to be resolvable [32] if there exists a dense subset Q of X such that X − Q is also dense in X .

Theorem 8.8

Let either every open subset of X be closed or ( X , τ , ℐ ) be resolvable. For a function f : ( X , τ , ℐ ) ⟶ ( Y , σ , J ) , then the following properties are equivalent:

1. f is a.e.c.;

2. f is a . p ℐ . c .;

3. f : ( X , 2 X ) ⟶ ( Y , σ , J ) is a . ℐ . c .

Proof

By Lemma 8.6, Theorem 8.7 and the former remarks, in either case, we have τ e = 2 X .□

Theorem 8.9

Let either X be a cofinite topological space, a countable topological space or the two point Sierpinski space. Then, for a function f : ( X , τ , ℐ ) ⟶ ( Y , σ , J ) , the following properties are equivalent:

1. f is a . δ β . c ;

2. f is a . s ℐ . c ;

3. f is a . ℐ . c .

Proof

This follows from Corollary 5.28 [3].□

Theorem 8.10

For a function f : ( X , τ , ℐ ) ⟶ ( Y , σ , J ) , the following properties are equivalent:

1. If every subset of X is nowhere dense and f is a.e.c., then f is a . s ℐ . c ;

2. If every subset of X is δ β ℐ -open and closed, then a.e.c. and a . δ β . c . are equivalent;

3. If X is an indiscrete space, then a.e.c. and a . p ℐ . c . are equivalent.

Proof

This follows from Proposition 2.2. of [3].□

Recall a topological space ( X , τ , ℐ ) is said to be semi-regular if for any open set Q of X and any x ∈ Q there exists a regular open set H of X such that x ∈ Q ⊆ H .

Theorem 8.11

If f : ( X , τ ) ⟶ ( Y , σ , J ) is an a.e.c. function and Y is semi-regular, then f is e - ℐ -continuous.

Proof

Let x ∈ X and Q be an open set of Y containing f ( x ) . According to semi-regularity of Y , there exists a regular open set K of Y such that f ( x ) ∈ K ⊆ Q . Since f is a.e.c., there exists H ∈ E O ℐ ( X , x ) such that f ( H ) ⊆ K . So, we have f ( H ) ⊆ K ⊆ Q . This shows that f is e - ℐ -continuous.□

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