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CUBO A Mathematical Journal
Vol.16, No¯ 01, (75–84). March 2015

Continuity via ΛsI-open sets

José Sanabria, Edumer Acosta, Ennis Rosas and Carlos Carpintero1

Departamento de Matemáticas, Núcleo de Sucre,

Universidad de Oriente,

Avenida Universidad, Cerro Colorado, Cumaná, Estado Sucre, Venezuela

jesanabri@gmail.com, edumeracostab@gmail.com, ennisrafael@gmail.com,

carpintero.carlos@gmail.com

ABSTRACT

Sanabria, Rosas and Carpintero [7] introduced the notions of ΛsI-sets and Λ
s
I-closed

sets using ideals on topological spaces. Given an ideal I on a topological space (X, τ),

a subset A ⊂ X is said to be ΛsI-closed if A = U ∩ F where U is a Λ
s
I-set and F is a

τ⋆-closed set. In this work we use sets that are complements of ΛsI-closed sets, which

are called ΛsI-open, to characterize new variants of continuity namely Λ
s
I-continuous,

quasi-ΛsI-continuous y Λ
s
I-irresolute functions.

RESUMEN

Sanabria, Rosas y Carpintero [7] introdujeron las nociones de conjuntos ΛsI y conjuntos

ΛsI-cerrados usando ideales sobre espacios topológicos. Dado un ideal I sobre un espacio

topológico (X, τ), un subconjunto A ⊂ X se llama ΛsI-cerrado si A = U ∩ F donde U es

un ΛsI-conjunto y F es un conjunto τ
⋆-cerrado . En este trabajo usamos conjuntos que

son complementos de conjuntos ΛsI-cerrado, los cuales son llamados Λ
s
I-abiertos, para

caracterizar nuevas variantes de continuidad, denominadas, funciones ΛsI-continuas y

funciones ΛsI-irresolutas.

Keywords and Phrases: Local function, ΛsI-open sets, Λ
s
I-irresolute functions.

2010 AMS Mathematics Subject Classification: 54C08, 54D05.

1Research Partially Suported by Consejo de Investigación UDO



76 José Sanabria, Edumer Acosta, Ennis Rosas and Carlos Carpintero CUBO
17, 1 (2015)

1 Introduction

The theory of ideal on topological spaces has been the subject of many studies in recent years.

It was the works of Hamlet and Jankovic [5], Abd El-Monsef, Lashien and Nasef [1] and Hatir

and Noiri [2] which motivated the research in applying topological ideals to generalize the most

basic properties in general topology. In 2002, Hatir and Noiri [2] introduced the notion of semi-

I-open sets in topological spaces. Also, Hatir and Noiri [3] investigated semi-I-open sets and

semi-I-continuous functions. Quite recently, Sanabria, Rosas and Carpintero [7] have introduced

the notions of ΛsI-sets and Λ
s
I-closed sets to obtain characterizations of two low separation axioms,

namely semi-I-T1 and semi-I-T1/2 spaces. In this article we introduce the notion of Λ
s
I-open sets

in order to characterize new variants of continuity in ideal topological spaces.

2 Preliminaries

Throughout this paper, P(X), Cl(A) and Int(A) denote the power set of X, the closure of A and

the interior of A, respectively. An ideal I on a topological space (X, τ) is a nonempty collection of

subsets of X which satisfies the following two properties:

(1) A ∈ I and B ⊂ A implies B ∈ I;

(2) A ∈ I and B ∈ I implies A ∪ B ∈ I.

A topological space (X, τ) with an ideal I on X is called an ideal topological space and is denoted by

(X, τ, I). Given an ideal topological space (X, τ, I), a set operator (.)⋆ : P(X) → P(X), called a local

function [6] of A with respect to τ and I, is defined as follows: for A ⊂ X, A⋆(I, τ) = {x ∈ X : U∩A /∈

Ifor every U ∈ τ(x)}, where τ(x) = {U ∈ τ : x ∈ U}. When there is no chance for confusion, we will

simply write A⋆ for A⋆(I, τ). In general, X⋆ is a proper subset of X. A Kuratowski closure operator

Cl⋆(.) for a topology τ⋆(I, τ), called the ⋆-topology, finer than τ, is defined by Cl⋆(A) = A∪A⋆(I, τ)

[5]. For any ideal topological space (X, τ, I), the collection β(I, τ) = {V \ J : V ∈ τ and J ∈ I} is

a basis for τ⋆(I, τ). According to the above, in this article, we denote by τ⋆ to topology τ⋆(I, τ)

generated by Cl⋆, that is, τ⋆ = {U ∈ P(X) : Cl⋆(X − U) = X − U}. The elements of τ⋆ are called

τ⋆-open and the complement of a τ⋆-open is called τ⋆-closed. It is well known that a subset A of

an ideal topological space (X, τ, I) is τ⋆-closed if and only if A⋆ ⊂ A [5].

Definition 2.1. A subset A of an ideal topological space (X, τ, I) is said to be semi-I-open [2] if

A ⊂ Cl
⋆

(Int(A)). The complement of a semi-I-open set is said to be semi-I-closed. The family of

all semi-I-open sets of an ideal topological space (X, τ, I) is denoted by SIO(X, τ).

The following three notions has been introduced by Sanabria et al. [7].

Definition 2.2. Let A be a subset of an ideal topological space (X, τ, I). A subset ΛsI(A) is defined

as follows: ΛsI(A) = ∩{U : A ⊂ U, U ∈ SIO(X, τ)}.



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Continuity via ΛsI-open sets 77

Definition 2.3. Let (X, τ, I) an ideal topological space. A subset A of X is said to be:

(1) ΛsI-set if A = Λ
s
I(A).

(2) ΛsI-closed if A = U ∩ F, where U is a Λ
s
I-set and F is an τ

⋆-closed set.

Since each open set is semi-I-open, by Propositions 3.1(3) and 4.1 of [7], we have the following

implications:

Open =⇒ semi-I-open =⇒ ΛsI-set =⇒ Λ
s
I-closed.

Lemma 2.1. (Sanabria, Rosas and Carpintero [7]) For an ideal topological space (X, τ, I), we take

τΛ
s
I = {A : A is a ΛsI-set of (X, τ, I)}. Then the pair (X, τ

Λ
s
I ) is an Alexandroff space.

Remark 2.1. According to Lemma 2.1, a subset A of an ideal topological space (X, τ, I) is open

in (X, τΛ
s
I ), if A is a ΛsI-set of (X, τ, I).

Definition 2.4. A subset A of an ideal topological space (X, τ, I) is called ΛsI-open if X \ A is a

ΛsI-closed set.

In the sequel, the ideal topological space (X, τ, I) is simply denoted by X. Next we present some

results related with ΛsI-open sets.

Lemma 2.2. Every τ⋆-open set is ΛsI-open.

Proof. This follows from Proposition 4.1 of [7].

Lemma 2.3. Let {Bα : α ∈ ∆} be a family of subsets of X. If Bα is Λ
s
I-open for each α ∈ ∆, then⋃

{Bα : α ∈ ∆} is Λ
s
I-open.

Proof. The proof is an immediate consequence from Proposition 4.2 of [7].

3 New variants of continuity

In this section we use the notions of open, ΛsI-open and τ
⋆-open sets in order to introduce new

forms of continuous functions called ΛsI-continuous, quasi-Λ
s
I-continuous and Λ

s
I-irresolute. We

study the relationships between these classes of functions and also obtain some properties and

characterizations of them.

Definition 3.1. A function f : (X, τ, I) → (Y, σ, J) is said to be semi-I-irresolute [4], if f−1(V) is

a semi-I-open set in (X, τ, I) for each semi-J-open set of (Y, σ, J).

Theorem 3.1. If a function f : (X, τ, I) → (Y, σ, J) is semi-I-irresolute, then f : (X, τΛ
s
I ) → (Y, σΛ

s
J )

is continuous.



78 José Sanabria, Edumer Acosta, Ennis Rosas and Carlos Carpintero CUBO
17, 1 (2015)

Proof. Let V be any ΛsJ-set of (Y, σ, J), that is V ∈ σ
Λ

s
J , then V = ΛsJ(V) = ∩{W : V ⊂

W and W is semi-J-open in (Y, σ, J)}. Since f is semi-I-irresolute, f−1(W) is a semi-I-open set in

(X, τ, I) for each W, hence we have

ΛsI(f
−1

(V)) = ∩{U : f−1(V) ⊂ U and U ∈ SIO(X, τ)}

⊂ ∩{f−1(W) : f−1(V) ⊂ f−1(W) and W ∈ SJO(Y, σ)}

= f−1(V).

On the other hand, always we have f−1(V) ⊂ ΛsI(f
−1(V)) and so f−1(V) = ΛsI(f

−1(V)). Therefore,

f−1(V) ∈ τΛ
s
I and f : (X, τΛ

s
I ) → (Y, σΛ

s
J ) is continuous.

Definition 3.2. A function f : (X, τ, I) → (Y, σ, J) is called:

(1) ΛsI-continuous, if f
−1(V) is a ΛsI-open set in (X, τ, I) for each open set V of (Y, σ, J).

(2) Quasi-ΛsI-continuous, if f
−1(V) is a ΛsI-open set in (X, τ, I) for each σ

⋆-open set V of (Y, σ, J).

(3) ΛsI-irresolute, if f
−1(V) is a ΛsI-open set in (X, τ, I) for each Λ

s
J-open set V of (Y, σ, J).

Theorem 3.2. If f : (X, τ, I) → (Y, σ, J) is ΛsI-irresolute function, then f is quasi-Λ
s
I-continuous.

Proof. Let V be a σ⋆-open set of (Y, σ, J), then by Lemma 2.2, we have V is a ΛsJ-open set of

(Y, σ, J) and since f is ΛsI-irresolute, f
−1(V) is a ΛsI-open set of (X, τ, I). Therefore, f is quasi-Λ

s
I-

continuous.

The following example shows a function quasi-ΛsI-continuous which is not Λ
s
I-irresolute.

Example 3.1. Let X = {a, b, c}, τ = {∅, {a, c}, X}, σ = {∅, {a}, {a, b}, {a, c}, X}, I = {∅, {c}} and

J = {∅, {b}}. The collection of the ΛsI-open sets of (X, τ, I) is {∅, {a, b}, {a, c}, {a}, {b}, X}, the collection

of the σ⋆-open sets of (X, σ, J) is {∅, {a}, {a, c}, {a, b}, X} and the collection of the ΛsJ-open sets of

(X, σ, J) is {{∅}, {a}, {b}, {c}, {a, c}, {b, c}, X}. The identity function f : (X, τ, I) → (X, σ, J) is quasi-ΛsI-

continuous, but is not ΛsI-irresolute, since f
−1({b, c}) = {b, c} and f−1({c}) = {c} are not ΛsI-open

sets.

Theorem 3.3. If f : (X, τ, I) → (Y, σ, J) is quasi-ΛsI-continuous function, then f is Λ
s
I-continuous.

Proof. Let V be an open set of (Y, σ, J), then V is σ⋆-open set of (Y, σ, J) and since f is quasi-ΛsI-

continuous, f−1(V) is a ΛsI-open set of (X, τ, I). This shows that f is Λ
s
I-continuous.

The following example shows a function ΛsI-continuous which is not quasi-Λ
s
I-continuous.

Example 3.2. Let X = {a, b, c}, τ = {∅, {a, c}, X}, σ = {∅, {a}, {b}, {a, b}, X}, I = {∅, {c}} and J =

{∅, {a}}. The collection of the ΛsI-open sets of (X, τ, I) is {∅, {a, b}, {a, c}, {a}, {b}, X} and the collection

of σ⋆-open sets of (X, σ, J) is {∅, {a}, {b, c}, {a, b}, {b}, X}. The identity function f : (X, τ, I) → (X, σ, J)

is ΛsI-continuous, but is not quasi-Λ
s
I-continuous, because f

−1({b, c}) = {b, c} is not a ΛsI-open set.



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Continuity via ΛsI-open sets 79

Corollary 3.1. If f : (X, τ, I) → (Y, σ, J) is a ΛsI-irresolute function, then f is Λ
s
I-continuous.

Proof. This is an immediate consequence of Theorems 3.2 and 3.3.

By the above results, we have the following diagram and none of these implications is reversible:

ΛsI-irresolute =⇒ quasi-Λ
s
I-continuous =⇒ Λ

s
I-continuous.

Proposition 3.1. Let f : (X, τ, I) → (Y, σ, J) and g : (Y, σ, J) → (Z, θ, K) be two functions, where

I, J, K are ideals on X, Y, Z respectively. Then:

(1) g ◦ f is ΛsI-irresolute, if f is Λ
s
I-irresolute and g is Λ

s
J-irresolute.

(2) g ◦ f is ΛsI-continuous, if f is Λ
s
I-irresolute and g is Λ

s
J-continuous.

(3) g ◦ f is ΛsI-continuous, if f is Λ
s
I-continuous and g is continuous.

(4) g ◦ f is quasi-ΛsI-continuous, if f is Λ
s
I-irresolute and g is quasi-Λ

s
J-continuous.

Proof. (1) Let V be a ΛsK-open set in (Z, θ, K). Since g is Λ
s
J-irresolute, then g

−1(V) is a ΛsJ-open

set in (Y, σ, J), using that f is ΛsI-irresolute, we obtain that f
−1(g−1(V)) is a ΛsI-open set in (X, τ, I).

But (g ◦ f)−1(V) = (f−1 ◦ g−1)(V) = f−1(g−1(V)) and hence, (g ◦ f)−1(V) is a ΛsI-open set in

(X, τ, I). This shows that g ◦ f is ΛsI-irresolute.

The proofs of (2), (3) and (4) are similar to the case (1).

In the next three theorems, we characterize ΛsI-continuous, quasi-Λ
s
I-continuous and Λ

s
I-irresolute

functions, respectively.

Theorem 3.4. For a function f : (X, τ, I) → (Y, σ), the following statements are equivalent:

(1) f is ΛsI-continuous.

(2) f−1(B) is a ΛsI-closed set in (X, τ, I) for each closed set B in (Y, σ).

(3) For each x ∈ X and each open set V in (Y, σ) containing f(x) there exists a ΛsI-open set U in

(X, τ, I) containing x such that f(U) ⊂ V.

Proof. (1)⇒(2) Let B be any closed set in (Y, σ), then V = Y\B is an open set in (Y, σ) and since f is

ΛsI-continuous, f
−1(V) is a ΛsI-open subset in (X, τ, I), but f

−1(V) = f−1(Y\B) = f−1(Y)\f−1(B) =

X \ f−1(B) and hence, f−1(B) is a ΛsI-closed set in (X, τ, I).

(2) ⇒ (1) Let V be any open set in (Y, σ), then B = Y \V is a closed set in (Y, σ). By hypothesis, we

have f−1(B) is a ΛsI-closed set in (X, τ, I), but f
−1(B) = f−1(Y \ V) = f−1(Y) \ f−1(V) = X \ f−1(V)

and so, f−1(V) is a ΛsI-open set in (X, τ, I). This shows that f is Λ
s
I-continuous.

(1) ⇒ (3) Let x ∈ X and V any open set in (Y, σ) such that f(x) ∈ V, then x ∈ f−1(V) and since f is

a ΛsI-continuous function, f
−1(V) is a ΛsI-open set in (X, τ, I). If U = f

−1(V), then U is a ΛsI-open



80 José Sanabria, Edumer Acosta, Ennis Rosas and Carlos Carpintero CUBO
17, 1 (2015)

set in (X, τ, I) containing x such that f(U) = f(f−1(V)) ⊂ V.

(3) ⇒ (1) Let V be any open set in (Y, σ) and x ∈ f−1(V), then f(x) ∈ V and by (3) there exists a

ΛsI-open set Ux in (X, τ, I) such that x ∈ Ux and f(Ux) ⊂ V. Thus, x ∈ Ux ⊂ f
−1(f(Ux)) ⊂ f

−1(V)

and hence f−1(V) =
⋃
{Ux : x ∈ f

−1(V)}. By Lemma 2.3, we have f−1(V) is a ΛsI-open set in

(X, τ, I) and so f is ΛsI-continuous.

Theorem 3.5. For a function f : (X, τ, I) → (Y, σ, J), the following statements are equivalent:

(1) f is quasi-ΛsI-continuous.

(2) f−1(B) is a ΛsI-closed set in (X, τ, I) for each σ
⋆-closed set B in (Y, σ, J).

(3) For each x ∈ X and each σ⋆-open set V in (Y, σ, J) containing f(x) there exists a ΛsI-open set

U in (X, τ, I) containing x such that f(U) ⊂ V.

Proof. The proof is similar to Theorem 3.4.

Theorem 3.6. For a function f : (X, τ, I) → (Y, σ, J), the following statements are equivalent:

(1) f is ΛsI-irresolute.

(2) f−1(B) is a ΛsI-closed set in (X, τ, I) for each Λ
s
J-closet set B in (Y, σ, J).

(3) For each x ∈ X and each ΛsJ-open set V in (Y, σ, J) containing f(x) there exists a Λ
s
I-open set

U in (X, τ, I) containing x such that f(U) ⊂ V.

Proof. The proof is similar to Theorem 3.4.

4 ΛsI-compactness and Λ
s
I-connectedness

In this section, new notions of compactness and connectedness are introduced in terms of ΛsI-open

sets and semi-I-open sets, in order to study their behavior under the direct images of the new

forms of continuity defined in the previous section.

Definition 4.1. An ideal topological space (X, τ, I) is said to be:

(1) ΛsI-compact if every cover of X by Λ
s
I-open sets has a finite subcover.

(2) τ⋆-compact if every cover of X by τ⋆-open sets has a finite subcover.

(3) Semi-I-compact if every cover of X by semi-I-open sets has a finite subcover.

Theorem 4.1. Let (X, τ, I) be an ideal topological space, the following properties hold:



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Continuity via ΛsI-open sets 81

(1) (X, τ, I) is ΛsI-compact if and only if for every collection {Aα : α ∈ ∆} of Λ
s
I-closed sets in

(X, τ, I) satisfying
⋂
{Aα : α ∈ ∆} = ∅, there is a finite subcollection Aα1, Aα2, . . . , Aαn with⋂

{Aαk : k = 1, . . . , n} = ∅.

(2) (X, τ, I) is τ⋆-compact if and only if for every collection {Aα : α ∈ ∆} of τ
⋆-closed sets in

(X, τ, I) satisfying
⋂
{Aα : α ∈ ∆} = ∅, there is a finite subcollection Aα1, Aα2, . . . , Aαn with⋂

{Aαk : k = 1, . . . , n} = ∅.

(3) (X, τ, I) is semi-I-compact if and only if for every collection {Aα : α ∈ ∆} of semi-I-closed sets

in (X, τ, I) satisfying
⋂
{Aα : α ∈ ∆} = ∅, there is a finite subcollection Aα1, Aα2, . . . , Aαn with⋂

{Aαk : k = 1, . . . , n} = ∅.

Proof. (1) Let {Aα : α ∈ ∆} be a collection of Λ
s
I-closed sets such that

⋂
{Aα : α ∈ ∆} = ∅, then

{X − Aα : α ∈ ∆} is a collection of Λ
s
I-open sets such that

X = X − ∅ = X −
⋂
{Aα : α ∈ ∆} =

⋃
{X − Aα : α ∈ ∆},

that is, {X − Aα : α ∈ ∆} is a cover of X by Λ
s
I-open sets. Since (X, τ, I) is Λ

s
I-compact, there exists

a finite subcollection X − Aα1, X − Aα2, . . . , X − Aαn such that

X =
⋃
{X − Aαk : k = 1, . . . , n} = X −

⋂
{Aαk : k = 1, . . . , n}.

This shows that
⋂
{Aαk : k = 1, . . . , n} = ∅. Conversely, suppose that {Uα : α ∈ ∆} is a cover of

X by ΛsI-open sets, then {X − Uα : α ∈ ∆} is a collection of Λ
s
I-closed sets such that

⋂
{X − Uα :

α ∈ ∆} = X −
⋃
{Uα : α ∈ ∆} = X − X = ∅. By hypothesis, there exists a finite subcollection

X − Uα1, X − Uα2, . . . , X − Uαn such that
⋂
{X − Uαk : k = 1, . . . , n} = ∅. Follows X = X − ∅ =

X −
⋂
{X − Uαk : k = 1, . . . , n} = X − (X −

⋃
{Uαk : k = 1, . . . , n}) =

⋃
{Uαk : k = 1, . . . , n}. This

shows that (X, τ, I) is ΛsI-compact.

The proofs of (2) and (3) are similar to the case (1).

Theorem 4.2. Let (X, τ, I) be an ideal topological space, the following properties hold:

(1) If (X, τΛ
s
I ) is compact, then (X, τ, I) is semi-I-compact.

(2) If (X, τ, I) is ΛsI-compact, then (X, τ, I) is τ
⋆-compact.

(3) If (X, τ, I) is ΛsI-compact, then (X, τ, I) is compact.

Proof. (1) Let {Uα : α ∈ ∆} any cover of X by semi-I-open sets, since every α ∈ ∆, Uα is a Λ
s
I-set

and hence, Uα ∈ τ
Λ

s
I for each α ∈ ∆. Since (X, τΛ

s
I ) is compact, there exists a finite subset ∆0 of

∆ such that X =
⋃
{Uα : α ∈ ∆0}. This shows that (X, τ) is semi-I-compact.

(2) Let {Fα : α ∈ ∆} be a collection of τ
⋆-closed sets of X such that

⋂
{Fα : α ∈ ∆} = ∅. Since

every τ⋆-closed set is ΛsI-closed, then {Fα : α ∈ ∆} is a collection of Λ
s
I-closed sets and (X, τ, I) is

ΛsI-compact. By Theorem 4.1(1), there exists a finite subset ∆0 of ∆ such that
⋂
{Fα : α ∈ ∆0} = ∅



82 José Sanabria, Edumer Acosta, Ennis Rosas and Carlos Carpintero CUBO
17, 1 (2015)

and by Theorem 4.1(2), we conclude that (X, τ, I) is τ⋆-compact.

(3) Follows from (2) and the fact that every τ⋆-compact space is compact.

Theorem 4.3. If f : (X, τ, I) → (Y, σ, J) is a surjective function, the following properties hold:

(1) If f is ΛsI-irresolute and (X, τ, I) is Λ
s
I-compact, then (Y, σ, J) is Λ

s
J-compact.

(2) If f is semi-I-irresolute and (X, τ, I) is semi-I-compact, then (Y, σ, J) is semi-J-compact.

(3) If f is quasi-ΛsI-continuous and (X, τ, I) is Λ
s
I-compact, then (Y, σ, J) is σ

⋆-compact.

(4) If f is ΛsI-continuous and (X, τ, I) is Λ
s
I-compact, then (Y, σ, J) is compact.

Proof. (1) Let {Vα : α ∈ ∆} be a cover of Y by Λ
s
J-open sets. Since f is Λ

s
I-irresolute, {f

−1(Vα) : α ∈

∆} is a cover of X by ΛsI-open sets and by the Λ
s
I-compactnes of (X, τ, I), there exists a finite subset

∆0 of ∆ such that X =
⋃
{f−1(Vα) : α ∈ ∆0}. Since f is surjective, then Y = f(X) = f(

⋃
{f−1(Vα) :

α ∈ ∆0}) =
⋃
{f(f−1(Vα)) : α ∈ ∆0} = {Vα : α ∈ ∆0} and this shows that (Y, θ, J) is Λ

s
J-compact.

The proofs of (2), (3) and (4) are similar to case (1).

Definition 4.2. An ideal topological space (X, τ, I) is said to be:

(1) ΛsI-connected if X cannot be written as a disjoint union of two nonempty Λ
s
I-open sets.

(2) τ⋆-connected if X cannot be written as a disjoint union of two nonempty τ⋆-open sets.

(3) semi-I-connected if X cannot be written as a disjoint union of two nonempty semi-I-open sets.

Theorem 4.4. Let (X, τ, I) be an ideal topological space, the following properties hold:

(1) If (X, τΛ
s
I ) is connected, then (X, τ; I) is semi-I-connected.

(2) If (X, τ, I) is ΛsI-connected, then (X, τ, I) is τ
⋆-connected.

(3) If (X, τ, I) is ΛsI-connected, then (X, τ, I) is connected.

Proof. (1) Suppose that (X, τ, I) is not semi-I-connected, then there exist non-empty semi-I-open

sets A and B such that A∩B = ∅ and A∪B = X. By Proposition 3.1(3) of [7], A and B are ΛsI-sets

and hence, (X, τΛ
s
I ) is not connected.

(2) Suppose that (X, τ, I) is not τ⋆-connected, then there exist non-empty τ⋆-open sets A and B

such that A ∩ B = ∅ and A ∪ B = X. By Lemma 2.2, we have A and B are ΛsI-open sets and so,

(X, τ, I) is not ΛsI-connected.

(3) Follows from (2) and the fact that every τ⋆-connected space is connected.

Theorem 4.5. For an ideal topological space (X, τ, I), the following statements are equivalent:

(1) (X, τ, I) is ΛsI-connected.



CUBO
17, 1 (2015)

Continuity via ΛsI-open sets 83

(2) ∅ and X are the only subsets of X which are both ΛsI-open and Λ
s
I-closed.

(3) Every ΛsI-continuous function of X into a discrete space Y with at least two points, is a constant

function.

Proof. (1)⇒(2) Let V be a subset of X which is both ΛsI-open and Λ
s
I-closed, then X − V is both

ΛsI-open and Λ
s
I-closed, so X = V ∪ (X − V). Since (X, τ, I) is Λ

s
I-connected, then one of those sets

is ∅. Therefore, V = ∅ or V = X.

(2)⇒(1) Suppose that (X, τ, I) is not ΛsI-connected and let X = U ∪ V, where U and V are disjoint

nonempty ΛsI-open sets in (X, τ, I), then U = X−V is both Λ
s
I-open and Λ

s
I-closed. By hypothesis,

U = ∅ or U = X, which is a contradiction. Therefore, (X, τ, I) is ΛsI-connected.

(2)⇒(3) Let f : (X, τ, I) → Y be a ΛsI-continuous function, where Y is a topological space with

the discrete topology and contains at least two points, then X can be cover by a collection of sets

which are both ΛsI-open and Λ
s
I-closed of the form {f

−1(y) : y ∈ Y}, from these, we conclude that

there exists a y0 ∈ Y such that f
−1({y0}) = X and so, f is a constant function.

(3)⇒(2) Let W be a subset of (X, τ, I) which is both ΛsI-open and Λ
s
I-closed. Suppose that W 6= ∅

and let f : (X, τ, I) → Y be the function defined by f(W) = {y1} and f(X − W) = {y2} for y1, y2 ∈ Y

and y1 6= y2. Then f is Λ
s
I-continuous, since the inverse image de each open set in Y is Λ

s
I-open

in X. Hence, by (3), f must be a constant function. It follows that X = W.

Theorem 4.6. If f : (X, τ, I) → (Y, σ, J) is a surjective function, the following properties hold:

(1) If f is a ΛsI-irresolute and (X, τ, I) is Λ
s
I-connected, then (Y, σ, J) is Λ

s
J-connected.

(2) If f is a semi-I-irresolute function and (X, τ, I) is semi-I-connected, then (Y, σ, J) is semi-J-

connected.

(3) If f is a quasi-ΛsI-continuous function and (X, τ, I) is Λ
s
I-connected, then (Y, σ, J) is σ

⋆-connected.

(4) If f is a ΛsI-continuous function and (X, τ, I) is Λ
s
I-connected, then (Y, σ) is connected.

Proof. (1) Suppose that (Y, σ, J) is not ΛsJ-connected, then there exist nonempty Λ
s
J-open sets

H, G in (Y, σ, J) such that G ∩ H = ∅ and G ∪ H = Y. Hence, we have f−1(G) ∩ f−1(H) = ∅,

f−1(G) ∪ f−1(H) = X and moreover, f−1(G) and f−1(H) are nonempty ΛsI-open sets in (X, τ, I).

This shows that (X, τ, I) is not ΛsI-connected.

The proofs of (2), (3) and (4) are similar to case (1).

Open problems. The Theorems 4.2 and 4.4 have been proved using the fact that every semi-

I-open set is ΛsI-open and that every τ
⋆-open set is ΛsI-open. But until today, we dont have any

contra example in order to shows that the converse of such Theorems are not true. In that sense

we write the following questions.

(1) Does there exists an ideal topological space (X, τ, I) which is semi-I-compact (resp. semi-I-

connected) but (X, τΛ
s
I ) is not a compact (resp. connected) space.?



84 José Sanabria, Edumer Acosta, Ennis Rosas and Carlos Carpintero CUBO
17, 1 (2015)

(2) Does there exists an ideal topological space (X, τ, I) which is τ⋆-compact (resp. τ⋆-connected)

but (X, τ) is not ΛsI-compact space (resp. Λ
s
I-connected space.)?

Received: January 2014. Accepted: January 2015.

References

[1] M. E. Abd El-Monsef, E. F. Lashien, A. A. Nasef: On I-open sets and I-continuous functions,

Kyungpook Math. J. 32 (1992), 21-30.

[2] E. Hatir, T. Noiri: On descompositions of continuity via idealization, Acta. Math. Hungar. 96

(4) (2002), 341-349.

[3] E. Hatir, T. Noiri: On semi-I-open sets and semi-I-continuous functions, Acta Math. Hungar.

107 (4) (2005), 345-353.

[4] E. Hatir, T. Noiri: On Hausdorff spaces via ideals and semi-I-irresolute Functions, Eur. J.

Pure. Appl. Math. 2 (2) (2009), 172-181.

[5] D. Jankovic, T. R. Hamlett: New topologies from old via ideals, Amer. Math. Monthly 97

(1990), 295-310.

[6] K. Kuratowski: Topology, Vol. I, Academic Press, New York-London, 1966.

[7] J. Sanabria, E. Rosas, C. Carpintero: On ΛsI-sets and the related notions in ideal topological

spaces, Math. Slovaca 63 (6) (2013), 1403-1411.


	Introduction
	Preliminaries
	New variants of continuity
	sI-compactness and sI-connectedness