On Faintly Continuous Functions via Generalized Topology

Abstract

The notion of faintly (𝜇,𝜎)-continuous function has been introduced. Relationship between this new class of function with similar types of functions has been given. Some characterizations and properties of such function are also being discussed.

1. Introduction

In topology, weak forms of open sets play an important role in the generalization of different forms of continuity. Using different forms of open sets, many authors have introduced and studied various types of continuity. In this paper, a unified version of some types of continuity has been introduced from a generalized topological space to a topological space. Generalized topology was first introduced by Császár (see [1–5]).

We recall some notions defined in [1]. Let 𝑋 be a nonempty set, and exp𝑋 denotes the power set of 𝑋. We call a class 𝜇⫅exp𝑋 a generalized topology [1], (GT) if Ø∈𝜇 and union of elements of 𝜇 belongs to 𝜇. A set 𝑋, with a GT 𝜇 on it, is said to be a generalized topological space (GTS) and is denoted by (𝑋,𝜇). Let (𝑋,𝜏) be a topological space. The 𝛿-closure [6] of a subset 𝐴 of a topological space (𝑋,𝜏) is defined by {𝑥∈𝑋∶𝐴∩𝑈≠Øforallregularopenset𝑈containing𝑥}, where a subset 𝐴 is called regular open if 𝐴=int(cl(𝐴)). A subset 𝐴 of a topological space (𝑋,𝜏) is called semiopen [7] (resp., preopen [8], 𝛼-open [9], 𝛽-open [10], 𝑏-open [11], 𝛿-preopen [12], 𝛿-semiopen [13], and 𝑒-open [14]) if 𝐴⫅cl(int(𝐴)) (resp., 𝐴⫅int(cl(𝐴)), 𝐴⫅int(cl(int(𝐴))), 𝐴⫅cl(int(cl𝐴)), 𝐴⫅cl(int(𝐴))∪int(cl(𝐴)), 𝐴⫅int(cl_𝛿(𝐴)), 𝐴⫅cl(int_𝛿(𝐴)), and 𝐴⫅int(cl_𝛿(𝐴))∪cl(int_𝛿(𝐴))). A point 𝑥∈𝑋 is in 𝑠cl(𝐴) (resp., 𝑝cl(𝐴)) if for each semiopen (resp., preopen) set 𝑈 containing 𝑥, 𝑈∩𝐴≠Ø. A point 𝑥∈𝑋 is called a 𝜃-cluster [6] (resp., semi-𝜃-cluster [15], 𝑝(𝜃)-cluster [16]) point of 𝐴 denoted by cl_𝜃(𝐴) (resp., 𝑠cl_𝜃(𝐴), 𝑝(𝜃)-cl(𝐴)) if cl(𝐴)∩𝑈≠Ø (resp., 𝑠cl(𝐴)∩𝑈≠Ø, 𝑝cl(𝐴)∩𝑈≠Ø) for every open (resp., semiopen, preopen) set 𝑈 containing 𝑥. A subset 𝐴 is called 𝜃-closed (resp., semi-𝜃-closed, 𝑝(𝜃)-closed) if cl_𝜃(𝐴)=𝐴 (resp., 𝑠cl_𝜃(𝐴)=𝐴, 𝑝(𝜃)-cl(𝐴)=𝐴). The complement of a 𝜃-closed (resp., semi-𝜃-closed, 𝑝(𝜃)-closed) set is called 𝜃-open (resp., semi-𝜃-open, 𝑝(𝜃)-open). The family of all 𝜃-open sets in a topological space forms a topology which is weaker than the original topology. The finite union of regular open sets is said to be 𝜋-open [17]. A subset 𝐴 of a topological space (𝑋,𝜏) is said to be 𝜋𝑔-closed [17] if cl(𝐴)⫅𝑈 whenever 𝐴⫅𝑈 and 𝑈 is 𝜋-open. A subset 𝐴 of 𝑋 is called 𝜔-open [18] if for each 𝑥∈𝐴 there exists an open set 𝑈 containing 𝑥 such that |𝑈⧵𝐴|≤ℵ₀. The family of all 𝜔-open subsets of a space (𝑋,𝜏) forms a topology on 𝑋 finer than 𝜏. For any topological space (𝑋,𝜏), the collection of all semiopen (resp., preopen, 𝛼-open, 𝛽-open, 𝑏-open, 𝛿-preopen, 𝛿-semiopen, 𝑒-open, 𝜃-open, semi-𝜃-open, 𝑝(𝜃)-open, 𝜋-open, and 𝜔-open) sets are denoted by 𝑆𝑂(𝑋) (resp., 𝑃𝑂(𝑋), 𝛼𝑂(𝑋), 𝛽𝑂(𝑋), 𝐵𝑂(𝑋), 𝛿𝑃𝑂(𝑋), 𝛿𝑆𝑂(𝑋), 𝑒𝑂(𝑋), 𝜃𝑂(𝑋), 𝑆_𝜃𝑂(𝑋), 𝑃_𝜃𝑂(𝑋), 𝜋𝑔𝑂(𝑋), and 𝜔𝑂(𝑋)). We note that each of these collections forms a GT on (𝑋,𝜏).

For a GTS (𝑋,𝜇), the elements of 𝜇 are called 𝜇-open sets and the complement of 𝜇-open sets are called 𝜇-closed sets. For 𝐴⫅𝑋, we denote by 𝑐_𝜇(𝐴) the intersection of all 𝜇-closed sets containing 𝐴, that is, the smallest 𝜇-closed set containing 𝐴, and by 𝑖_𝜇(𝐴) the union of all 𝜇-open sets contained in 𝐴, that is, the largest 𝜇-open set contained in 𝐴 (see [1, 2]).

It is easy to observe that 𝑖_𝜇 and 𝑐_𝜇 are idempotent and monotonic, where 𝛾: exp𝑋→exp𝑋 is said to be idempotent if and only if 𝐴⫅𝐵⫅𝑋 implies 𝛾(𝛾(𝐴))=𝛾(𝐴) and monotonic if and only if 𝛾(𝐴)⫅𝛾(𝐵). It is also well known from [2, 3] that if 𝜇 is a GT on 𝑋 and 𝐴⫅𝑋, 𝑥∈𝑋, then 𝑥∈𝑐_𝜇(𝐴) if and only if 𝑥∈𝑀∈𝜇 ⇒ 𝑀∩𝐴≠Ø and 𝑐_𝜇(𝑋⧵𝐴)=𝑋⧵𝑖_𝜇(𝐴).

Hereafter, throughout the paper, we will use (𝑋,𝜇) to mean a generalized topological space and (𝑌,𝜎) to be a topological space unless otherwise stated.

2. Faintly (𝜇,𝜎)-Continuous and Related Functions

Definition 1. A function 𝑓∶(𝑋,𝜇)→(𝑌,𝜎) is said to be faintly (𝜇,𝜎)-continuous at 𝑥 if for each 𝜃-open set 𝑉 in 𝑌 containing 𝑓(𝑥) there exists 𝑈 in 𝜇 containing 𝑥 such that 𝑓(𝑈)⫅𝑉. If 𝑓 is faintly (𝜇,𝜎)-continuous at each point of 𝑋, then 𝑓 is called faintly (𝜇,𝜎)-continuous on 𝑋.

Definition 2. A function 𝑓∶(𝑋,𝜇)→(𝑌,𝜎) is said to be (𝜇,𝜎)-continuous [19] (resp., weakly (𝜇,𝜎)-continuous, almost (𝜇,𝜎)-continuous) if for each 𝑥∈𝑋 and each open set 𝑉 of 𝑌 containing 𝑓(𝑥) there exists 𝑈∈𝜇 containing 𝑥 such that 𝑓(𝑈)⫅𝑉 (resp., 𝑓(𝑈)⫅cl(𝑉), 𝑓(𝑈)⫅int(cl(𝑉))).

Remark 3. (i) Let (𝑋,𝜏) and (𝑌,𝜎) be two topological spaces. If 𝜇=𝜏, 𝑆𝑂(𝑋), 𝑃𝑂(𝑋), 𝛼𝑂(𝑋), 𝛽𝑂(𝑋), 𝐵𝑂(𝑋), 𝛿𝑃𝑂(𝑋), 𝛿𝑆𝑂(𝑋), 𝑒𝑂(𝑋), 𝜃𝑂(𝑋), 𝑆_𝜃𝑂(𝑋), 𝑃_𝜃𝑂(𝑋), 𝜋𝑔𝑂(𝑋), and 𝜔𝑂(𝑋), then a faintly (𝜇,𝜎)-continuous function 𝑓∶(𝑋,𝜇)→(𝑌,𝜎) reduces to a faintly continuous [20], faintly semicontinuous [21], faintly precontinuous [21], faintly 𝛼-continuous [22, 23], faintly 𝛽-continuous [21], faintly 𝛾-continuous [21], faintly 𝛿-precontinuous [24], faintly 𝛿-semicontinuous [25], faintly 𝑒-continuous [26], quasi-𝜃-continuous [21], faintly semi-𝜃 continuous [27], faintly pre-𝜃-continuous [27], faintly 𝜋𝑔-continuous function [28], and faintly 𝜔-continuous function [29], respectively. On the other hand, every faintly 𝑚-continuous function [30] 𝑓∶(𝑋,𝑚_𝑋)→(𝑌,𝜎) is faintly (𝑚_𝑋,𝜎)-continuous if 𝑚_𝑋 is closed under arbitrary union (as 𝑚_𝑋 is a GT in that case).
(ii) It follows from Definitions 1 and 2 that(a)every (𝜇,𝜎)-continuous function is faintly (𝜇,𝜎)-continuous;(b)every (𝜇,𝜎)-continuous function is almost (𝜇,𝜎)-continuous;(c)every almost (𝜇,𝜎)-continuous function is weakly (𝜇,𝜎)-continuous.
But the converses are false as shown in the next example.

Example 4. (a) Let 𝑋=𝑌={𝑎,𝑏,𝑐}, let 𝜇={Ø,{𝑎,𝑏},{𝑏,𝑐},𝑋}, and let 𝜎={Ø,{𝑎},{𝑎,𝑏},𝑋}. Consider the function 𝑓∶(𝑋,𝜇)→(𝑌,𝜎) defined by 𝑓(𝑎)=𝑎,𝑓(𝑏)=𝑏, and 𝑓(𝑐)=𝑎. It is easy to check that 𝑓 is faintly (𝜇,𝜎)-continuous but not (𝜇,𝜎)-continuous.
(b) Let 𝑋=𝑌={𝑎,𝑏,𝑐}, let 𝜇={Ø,{𝑎},{𝑎,𝑏},{𝑏,𝑐},𝑋}, and let 𝜎={Ø,{𝑎},{𝑐},{𝑎,𝑐},{𝑏,𝑐},𝑋}. Consider the identity map 𝑓∶(𝑋,𝜇)→(𝑌,𝜎). It can be easily checked that 𝑓 is almost (𝜇,𝜎)-continuous but not (𝜇,𝜎)-continuous.
(c) Let 𝑋=𝑌={𝑎,𝑏,𝑐}, let 𝜇={Ø,{𝑎},{𝑎,𝑏},{𝑏,𝑐},𝑋}, and let 𝜎={Ø,{𝑎},{𝑐},{𝑎,𝑐},𝑋}. Consider the identity map 𝑓∶(𝑋,𝜇)→(𝑌,𝜎). It can be easily checked that 𝑓 is not almost (𝜇,𝜎)-continuous but weakly (𝜇,𝜎)-continuous.

Theorem 5. For a function 𝑓∶(𝑋,𝜇)→(𝑌,𝜎), the following statements are equivalent: (a)𝑓 is faintly (𝜇,𝜎)-continuous;(b)𝑓⁻¹(𝑉) is 𝜇-open for each 𝜃-open set 𝑉 of 𝑌;(c)𝑓⁻¹(𝐹) is 𝜇-closed for each 𝜃-closed set 𝐹 of 𝑌.

Proof. (a)⇒(b): Let 𝑉 be a 𝜃-open set of 𝑌, and let 𝑥∈𝑓⁻¹(𝑉). Since 𝑓(𝑥)∈𝑉 and 𝑓 is faintly (𝜇,𝜎)-continuous, there exists 𝑈∈𝜇 containing 𝑥 such that 𝑓(𝑈)⫅𝑉. It then follows that 𝑥∈𝑈⫅𝑓⁻¹(𝑉). Hence, 𝑓⁻¹(𝑉) is 𝜇-open.
(b)⇒(a): Let 𝑥∈𝑋 and let 𝑉 be a 𝜃-open set in 𝑌 containing 𝑓(𝑥). Then, by (b), 𝑓⁻¹(𝑉) is 𝜇-open in 𝑋 containing 𝑥. Let 𝑈=𝑓⁻¹(𝑉). Then, 𝑓(𝑈)⫅𝑉.
(b)⇒(c): Let 𝐹 be a 𝜃-closed set of 𝑌. Since 𝑌⧵𝐹 is 𝜃-open, by (b), it follows that 𝑓⁻¹(𝑌⧵𝐹)=𝑋⧵𝑓⁻¹(𝐹) is 𝜇-open. This shows that 𝑓⁻¹(𝐹) is 𝜃-closed.
(c)⇒(b): Let 𝑉 be a 𝜃-open set in 𝑌. Then, 𝑌⧵𝑉 is 𝜃-closed in 𝑌. By (c), 𝑓⁻¹(𝑌⧵𝑉)=𝑋⧵𝑓⁻¹(𝑉) is 𝜃-closed. Thus, 𝑓⁻¹(𝑉) is 𝜇-open.

Theorem 6. If a function 𝑓∶(𝑋,𝜇)→(𝑌,𝜎) is weakly (𝜇,𝜎)-continuous, then it is faintly (𝜇,𝜎)-continuous.

Proof. Let 𝑥∈𝑋, and let 𝑉 be a 𝜃-open set containing 𝑓(𝑥). Then, there exists an open set 𝑊 such that 𝑓(𝑥)∈𝑊⫅cl(𝑊)⫅𝑉. Since 𝑓 is weakly (𝜇,𝜎)-continuous, there exists a 𝜇-open set 𝑈 containing 𝑥 such that 𝑓(𝑈)⫅cl(𝑊)⫅𝑉. Thus, 𝑓 is faintly (𝜇,𝜎)-continuous.

Example 7. Let 𝑋=𝑌={𝑎,𝑏,𝑐}, let 𝜇={Ø,{𝑎},{𝑎,𝑏},𝑋}, and let 𝜎={Ø,{𝑎},{𝑏},{𝑎,𝑏},𝑋}. The identity function 𝑓∶(𝑋,𝜇)→(𝑌,𝜎) is faintly (𝜇,𝜎)-continuous but not weakly (𝜇,𝜎)-continuous.

Definition 8. A function 𝑓∶(𝑋,𝜇)→(𝑌,𝜎) is called slightly (𝜇,𝜎)-continuous if for each 𝑥∈𝑋 and each clopen set 𝑉 of 𝑌 containing 𝑓(𝑥) there exists a 𝜇-open set 𝑈 containing 𝑥 such that 𝑓(𝑈)⫅𝑉.

Theorem 9. If 𝑓∶(𝑋,𝜇)→(𝑌,𝜎) is faintly (𝜇,𝜎)-continuous, then it is slightly (𝜇,𝜎)-continuous.

Proof. Let 𝑥∈𝑋, and let 𝑉 be a clopen set containing 𝑓(𝑥). Then, 𝑉 is 𝜃-open. Since 𝑓 is faintly (𝜇,𝜎)-continuous, there exists 𝑈∈𝜇 containing 𝑥 such that 𝑓(𝑈)⫅𝑉 showing 𝑓 to be slightly (𝜇,𝜎)-continuous.

Example 10. Let ℝ be the set of real numbers. Consider the identity mapping 𝑓∶(ℝ,𝜏_co)→(ℝ,𝜏_𝑢), where 𝜏_co and 𝜏_𝑢 denote the cocountable and usual topology, respectively. It is easy to show that 𝑓 is slightly (𝜇,𝜎)-continuous but not faintly (𝜇,𝜎)-continuous.

Remark 11. From Remark 3 and Theorems 6 and 9, we have the following implications:(𝜇,𝜎)-continuity ⇒ almost (𝜇,𝜎)-continuity ⇒ weakly (𝜇,𝜎)-continuity ⇒ faintly (𝜇,𝜎)-continuity ⇒ slightly (𝜇,𝜎)-continuity.

Theorem 12. Let (𝑌,𝜎) be regular. If a function 𝑓∶(𝑋,𝜇)→(𝑌,𝜎) is faintly (𝜇,𝜎)-continuous, then it is (𝜇,𝜎)-continuous.

Proof. Let 𝑉 be an open set of 𝑌 containing 𝑓(𝑥). Since 𝑌 is regular, 𝑉 is 𝜃-open in 𝑌. Then by Theorem 5, 𝑓⁻¹(𝑉) is 𝜇-open in 𝑋. Let 𝑈=𝑓⁻¹(𝑉). Then, 𝑈 is a 𝜇-open set containing 𝑥 such that 𝑓(𝑈)⫅𝑉. Thus, 𝑓 is (𝜇,𝜎)-continuous.

Definition 13. A topological space (𝑌,𝜎) is said to be almost regular [31] if for any regular closed set 𝐹 and any point 𝑥∈𝑌⧵𝐹 there exist disjoint open sets 𝑈 and 𝑉 such that 𝑥∈𝑈 and 𝐹⫅𝑉.

Theorem 14. If a function 𝑓∶(𝑋,𝜇)→(𝑌,𝜎) is faintly (𝜇,𝜎)-continuous and (𝑌,𝜎) is almost regular, then 𝑓 is almost (𝜇,𝜎)-continuous.

Proof. Let 𝑥∈𝑋, and let 𝑉 be an open set in 𝑌. Then, int(cl(𝑉)) is a regular open set in 𝑌 so that it is 𝜃-open in 𝑌 (as in an almost regular space, every regular open set is 𝜃-open [20]). Hence, by faintly (𝜇,𝜎)-continuity of 𝑓, there exists a 𝜇-open set 𝑈 containing 𝑥 such that 𝑓(𝑈)⫅int(cl(𝑉)) showing 𝑓 to be almost (𝜇,𝜎)-continuous.

The clopen subsets of a topological space (𝑋,𝜏) form a base for a topology on 𝑋. This topology is called ultraregularization [32] of 𝜏 and is denoted by 𝜏_𝑢. A topological space (𝑋,𝜏) is said to be ultra-regular [33] if 𝜏=𝜏_𝑢.

Theorem 15. If (𝑌,𝜎) is ultraregular, then for a function 𝑓∶(𝑋,𝜇)→(𝑌,𝜎) the following are equivalent:(i)𝑓 is (𝜇,𝜎)-continuous;(ii)𝑓 is almost (𝜇,𝜎)-continuous;(iii)𝑓 is weakly (𝜇,𝜎)-continuous;(iv)𝑓 is faintly (𝜇,𝜎)-continuous;(v)𝑓 is slightly (𝜇,𝜎)-continuous.

Proof. We will only show that in an ultra regular space every slightly (𝜇,𝜎)-continuous function is (𝜇,𝜎)-continuous. The rest will follow from Remark 11. Let 𝑉 be an open set in 𝑌 containing 𝑓(𝑥). Then, as (𝑌,𝜎) is ultra regular, there exists a clopen set 𝑊 in 𝑌 containing 𝑓(𝑥) such that 𝑊⫅𝑉. Since 𝑓 is slightly (𝜇,𝜎)-continuous, there exists a 𝜇-open set 𝑈 in 𝑋 containing 𝑥 such that 𝑓(𝑈)⫅𝑊⫅𝑉. Thus, 𝑓 is (𝜇,𝜎)-continuous.

Theorem 16. If 𝜇 and 𝜆 are two GTs on 𝑋 such that 𝜇⫅𝜆 and if 𝑓∶(𝑋,𝜇)→(𝑌,𝜎) is faintly (𝜇,𝜎)-continuous, then 𝑓 is faintly (𝜆,𝜎)-continuous.

Proof. It follows immediately from Theorem 5.

Observation. Let (𝑋,𝜏) be a topological space. Then, we have(i)𝜏⫅𝛼𝑂(𝑋)⫅𝑃𝑂(𝑋)⫅𝐵𝑂(𝑋)⫅𝛽𝑂(𝑋); (ii)𝜏⫅𝛼𝑂(𝑋)⫅𝑃𝑂(𝑋)⫅𝛿𝑃𝑂(𝑋)⫅𝑒𝑂(𝑋); (iii)𝜏⫅𝛼𝑂(𝑋)⫅𝑆𝑂(𝑋)⫅𝐵𝑂(𝑋)⫅𝛽𝑂(𝑋); (iv)𝜏⫅𝛼𝑂(𝑋)⫅𝑆𝑂(𝑋)⫅𝑒𝑂(𝑋); (v)𝑃_𝜃𝑂(𝑋)⫅𝑃𝑂(𝑋), 𝜃𝑂(𝑋)⫅𝑆_𝜃𝑂(𝑋)⫅𝛿𝑆𝑂(𝑋)⫅𝑆𝑂(𝑋). Thus, from Theorem 16, we can deduce relations between different types of faintly (𝜇,𝜎)-continuous functions.

3. Properties of Faintly (𝜇,𝜎)-Continuous Functions

Definition 17. A GTS (𝑋,𝜇) (resp., a topological space (𝑋,𝜏)) is said to be 𝜇-𝑇₂ [34] (resp., 𝜃-𝑇₂ [35]) if for any two distinct points 𝑥, 𝑦 of 𝑋 there exist two disjoint 𝜇-open (resp., 𝜃-open) sets 𝑈 and 𝑉 containing 𝑥 and 𝑦, respectively.
It is well known from [36] that a topological space (𝑋,𝜏) is Hausdorff if and only if it is 𝜃-𝑇₂.

Theorem 18. If 𝑓∶(𝑋,𝜇)→(𝑌,𝜎) is a faintly (𝜇,𝜎)-continuous injection and (𝑌,𝜎) is 𝑇₂, then (𝑋,𝜇) is 𝜇-𝑇₂.

Proof. Let 𝑥 and 𝑦 be two distinct points of 𝑋. Then, 𝑓(𝑥) and 𝑓(𝑦) are two distinct points of 𝑌. Thus, there exist two disjoint open sets 𝑈 and 𝑉 containing 𝑥 and 𝑦 respectively. Then, by Theorem 5, 𝑓⁻¹(𝑈) and 𝑓⁻¹(𝑉) are two 𝜇-open sets in 𝑋. Clearly, 𝑥∈𝑓⁻¹(𝑈), 𝑦∈𝑓⁻¹(𝑉), and 𝑓⁻¹(𝑈)∩𝑓⁻¹(𝑉)=Ø showing (𝑋,𝜇) to be 𝜇-𝑇₂.

Definition 19. A topological space (𝑋,𝜏) is said to be(i)𝜃-regular [26] if for each 𝜃-closed set 𝐹 and each point 𝑥∉𝐹 there exist disjoint 𝜃-open sets 𝑈 and 𝑉 such that 𝑥∈𝑈, 𝐹⫅𝑉;(ii)𝜃-normal [26] if for any two disjoint 𝜃-closed subsets 𝐹 and 𝐾 there exist disjoint 𝜃-open sets 𝑈 and 𝑉 such that 𝐹⫅𝑈, 𝐾⫅𝑉.

Definition 20. A GTS (𝑋,𝜇) is said to be(i)𝜇-regular [37] if for each 𝜇-closed set 𝐹 and each point 𝑥∉𝐹, there exist disjoint 𝜇-open sets 𝑈 and 𝑉 such that 𝑥∈𝑈, 𝐹⫅𝑉.(ii)𝜇-normal [37] if for any two disjoint 𝜇-closed subsets 𝐹 and 𝐾, there exist disjoint 𝜇-open sets 𝑈 and 𝑉 such that 𝐹⫅𝑈, 𝐾⫅𝑉.

Definition 21. A function 𝑓∶(𝑋,𝜇)→(𝑌,𝜎) is called (𝜇,𝜎_𝜃)-open if for each 𝜇-open set 𝑉 in 𝑋,𝑓(𝑉) is 𝜃-open in (𝑌,𝜎).

Theorem 22. If 𝑓∶(𝑋,𝜇)→(𝑌,𝜎) is faintly (𝜇,𝜎)-continuous, (𝜇,𝜎_𝜃)-open bijection, and (𝑋,𝜇) is 𝜇-regular, then (𝑌,𝜎) is 𝜃-regular.

Proof. Let 𝐹 be a 𝜃-closed subset of 𝑌, and let 𝑦∉𝐹. Let 𝑦=𝑓(𝑥). Since 𝑓 is faintly (𝜇,𝜎)-continuous, by Theorem 5, 𝑓⁻¹(𝐹) is 𝜇-closed in 𝑋 so that 𝑓⁻¹(𝑦)=𝑥∉𝑓⁻¹(𝐹). Let 𝐺=𝑓⁻¹(𝐹). Then, 𝑥∉𝐺; thus, by 𝜇-regularity of (𝑋,𝜇), there exist two disjoint 𝜇-open sets 𝑈 and 𝑉 such that 𝐺⫅𝑈 and 𝑥∈𝑉. Thus, we have 𝐹=𝑓(𝐺)⫅𝑓(𝑈) and 𝑦=𝑓(𝑥)∈𝑓(𝑉) and 𝑓(𝑈)∩𝑓(𝑉)=Ø. As 𝑓 is (𝜇,𝜎_𝜃)-open, 𝑓(𝑈) and 𝑓(𝑉) are 𝜃-open in (𝑌,𝜎) showing (𝑌,𝜎) to be 𝜃-regular.

Theorem 23. If 𝑓∶(𝑋,𝜇)→(𝑌,𝜎) is faintly (𝜇,𝜎)-continuous, (𝜇,𝜎_𝜃)-open surjection, and (𝑋,𝜇) is 𝜇-normal, then (𝑌,𝜎) is 𝜃-normal.

Proof. Let 𝐹₁ and 𝐹₂ be two disjoint 𝜃-closed subsets of (𝑌,𝜎). Since 𝑓 is faintly (𝜇,𝜎)-continuous, by Theorem 5, 𝑓⁻¹(𝐹₁) and 𝑓⁻¹(𝐹₂) are two 𝜇-closed subsets in 𝑋. Let 𝐾₁=𝑓⁻¹(𝐹₁), and let 𝐾₂=𝑓⁻¹(𝐹₂). Then, 𝐾₁ and 𝐾₂ are two disjoint 𝜇-closed subsets of (𝑋,𝜇). Since (𝑋,𝜇) is 𝜇-normal, there exist two disjoint 𝜇-open sets 𝑈 and 𝑉 such that 𝐾₁⫅𝑈 and 𝐾₂⫅𝑉. We thus have 𝐹₁=𝑓(𝐾₁)⫅𝑓(𝑈) and 𝐹₂=𝑓(𝐾₂)⫅𝑓(𝑉). Also 𝑓(𝑈) and 𝑓(𝑉) are two disjoint 𝜃-open sets in (𝑌,𝜎) showing (𝑌,𝜎) to be 𝜃-normal.

Definition 24. A GTS (𝑋,𝜇) is said to be 𝜇-connected [38] if 𝑋 cannot be written as union of two nonempty 𝜇-open sets.

Theorem 25. If 𝑓∶(𝑋,𝜇)→(𝑌,𝜎) is a faintly (𝜇,𝜎)-continuous surjection and (𝑋,𝜇) is 𝜇-connected, then (𝑌,𝜎) is connected.

Proof. Let us assume that (𝑌,𝜎) be not connected. Then, there exist nonempty disjoint open sets 𝑉₁ and 𝑉₂ such that 𝑌=𝑉₁∪𝑉₂. Hence, we have 𝑓⁻¹(𝑉₁)∪𝑓⁻¹(𝑉₂)=𝑋 and 𝑓⁻¹(𝑉₁)∩𝑓⁻¹(𝑉₂)=Ø. Since 𝑓 is surjective, 𝑓⁻¹(𝑉₁) and 𝑓⁻¹(𝑉₂) are nonempty. Since 𝑉₁ and 𝑉₂ are clopen, they are 𝜃-open in 𝑌. Thus by Theorem 5, 𝑓⁻¹(𝑉₁) and 𝑓⁻¹(𝑉₂) are 𝜇-open. Therefore (𝑋,𝜇) is not 𝜇-connected—a contradiction.

Definition 26. A GTS (𝑋,𝜇) is called 𝜇-compact [39] if every 𝜇-open cover of 𝑋 has a finite subcover. A subset 𝐴 of 𝑋 is said to be 𝜇-compact relative to (𝑋,𝜇) if every cover of 𝐴 by 𝜇-open sets of 𝑋 has a finite subcover.

Definition 27. A subset 𝐴 of a topological space (𝑌,𝜎) is called 𝜃-compact relative to (𝑌,𝜎) [40] if every cover of 𝐴 by 𝜃-open sets of (𝑌,𝜎) has a finite subcover. A topological space (𝑌,𝜎) is called 𝜃-compact if 𝑌 is 𝜃-compact relative to (𝑌,𝜎).

Theorem 28. If 𝑓∶(𝑋,𝜇)→(𝑌,𝜎) is a faintly (𝜇,𝜎)-continuous function and 𝐴 is 𝜇-compact relative to (𝑋,𝜇), then 𝑓(𝐴) is 𝜃-compact relative to (𝑌,𝜎).

Proof. Let {𝑉_𝛼∶𝛼∈Λ} be a cover of 𝑓(𝐴) by 𝜃-open sets of 𝑌. Then, for each 𝑥∈𝐴, there exists 𝛼_𝑥∈Λ such 𝑓(𝑥)∈𝑉_𝛼𝑥. Since 𝑓 is a faintly (𝜇,𝜎)-continuous function, there exists 𝑈_𝑥∈𝜇 containing 𝑥 such that 𝑓(𝑈_𝑥)⫅𝑉_𝛼𝑥. Then, the family {𝑈_𝑥∶𝑥∈𝐴} is a cover of 𝐴 by 𝜇-open sets of (𝑋,𝜇). Since 𝐴 is 𝜇-compact relative to (𝑋,𝜇), there exists a finite number of points, say, 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝐴, such that 𝐴⫅∪{𝑈_𝑥𝑖∶𝑖=1,2,…,𝑛,𝑥_𝑖∈𝐴}. Therefore, we have 𝑓(𝐴)⫅∪{𝑓(𝑈_𝑥𝑖)∶𝑖=1,2,…,𝑛,𝑥_𝑖∈𝐴}⫅∪{𝑉_{𝛼𝑖)(𝑥}∶𝑖=1,2,…,𝑛,𝑥_𝑖∈𝐴}. This shows that 𝑓(𝐴) is 𝜃-compact relative to (𝑌,𝜎).

Definition 29. For any subset 𝐴 of a GTS (𝑋,𝜇), the 𝜇-frontier of 𝐴 is denoted by Fr_𝜇(𝐴) and defined by Fr_𝜇(𝐴)=𝑐_𝜇(𝐴)∩𝑐_𝜇(𝑋⧵𝐴).

Theorem 30. The set of all points 𝑥∈𝑋 at which the function 𝑓∶(𝑋,𝜇)→(𝑌,𝜎) is not faintly (𝜇,𝜎)-continuous is identical with the union of 𝜇-frontier of the inverse images of 𝜃-open sets of (𝑌,𝜎) containing 𝑓(𝑥).

Proof. Suppose that 𝑓 is not faintly (𝜇,𝜎)-continuous at 𝑥∈𝑋. Then, there exists a 𝜃-open set 𝑉 of 𝑌 containing 𝑓(𝑥) such that 𝑓(𝑈) is not a subset of 𝑉 for each 𝜇-open set 𝑈 containing 𝑥∈𝑋. Hence, we have 𝑈∩(𝑋⧵𝑓⁻¹(𝑉))≠Ø for each 𝑈∈𝜇 containing 𝑥. So, 𝑥∈𝑐_𝜇(𝑋⧵𝑓⁻¹(𝑉)). On the other hand, we have 𝑥∈𝑓⁻¹(𝑉)⫅𝑐_𝜇(𝑓⁻¹(𝑉)). Hence 𝑥∈Fr_𝜇(𝑓⁻¹(𝑉)).
Conversely, suppose that 𝑓 is not faintly (𝜇,𝜎)-continuous at 𝑥∈𝑋, and let 𝑉 be any 𝜃-open set containing 𝑓(𝑥). Then, by Theorem 5, 𝑓⁻¹(𝑉)=𝑖_𝜇(𝑓⁻¹(𝑉)). Therefore, 𝑥∉Fr_𝜇(𝑓⁻¹(𝑉)) for each 𝜃-open set 𝑉 containing 𝑓(𝑥). This completes the proof.

4. Faintly 𝜇-Closed Graph

Definition 31. A function 𝑓∶(𝑋,𝜇)→(𝑌,𝜎) is said to have a faintly 𝜇-closed graph if for each (𝑥,𝑦)∈(𝑋×𝑌)⧵𝐺(𝑓) there exist 𝑈∈𝜇 containing 𝑥 and a 𝜃-open set 𝑉 in 𝑌 containing 𝑦 such that (𝑈×𝑉)∩𝐺(𝑓)=Ø.

Lemma 32. The graph 𝐺(𝑓) of a function 𝑓∶(𝑋,𝜇)→(𝑌,𝜎) is faintly 𝜇-closed if and only if for each (𝑥,𝑦)∈(𝑋×𝑌)⧵𝐺(𝑓) there exist 𝑈∈𝜇 containing 𝑥 and a 𝜃-open set 𝑉 in 𝑌 containing 𝑦 such that 𝑓(𝑈)∩𝑉=Ø.

Theorem 33. Let a function 𝑓∶(𝑋,𝜇)→(𝑌,𝜎) have a faintly 𝜇-closed graph 𝐺(𝑓). If 𝑓 is a faintly (𝜇,𝜎)-continuous injection, then (𝑋,𝜇) is 𝜇-𝑇₂.

Proof. Let 𝑥 and 𝑦 be two distinct points of 𝑋. Then, since 𝑓 is an injection, we have 𝑓(𝑥)≠𝑓(𝑦). Then, we have (𝑥,𝑓(𝑦))∈𝑋×𝑌⧵𝐺(𝑓). Thus, by Lemma 32, there exist a 𝜇-open set 𝑈 containing 𝑥 in 𝑋 and a 𝜃-open set 𝑉 in 𝑌 containing 𝑓(𝑦) such that 𝑓(𝑈)∩𝑉=Ø. Hence, 𝑈∩𝑓⁻¹(𝑉)=Ø, and 𝑦∉𝑈. Since 𝑓 is faintly (𝜇,𝜎)-continuous, there exists a 𝜇-open set 𝑊 containing 𝑦 such that 𝑓(𝑊)⫅𝑉. Therefore, we have 𝑓(𝑈)∩𝑓(𝑊)=Ø. Since 𝑓 is injective, we obtain 𝑈∩𝑊=Ø showing (𝑋,𝜇) to be 𝜇-𝑇₂.

Definition 34. A function 𝑓∶(𝑋,𝜇)→(𝑌,𝜎) is said to have a faintly strong 𝜇-closed graph if for each (𝑥,𝑦)∈(𝑋×𝑌)⧵𝐺(𝑓) there exist 𝑈∈𝜇 containing 𝑥 and an open set 𝑉 in 𝑌 containing 𝑦 such that (𝑈×cl(𝑉))∩𝐺(𝑓)=Ø.

Lemma 35. The graph 𝐺(𝑓) of a function 𝑓∶(𝑋,𝜇)→(𝑌,𝜎) is faintly strong 𝜇-closed if and only if for each (𝑥,𝑦)∈(𝑋×𝑌)⧵𝐺(𝑓) there exist 𝑈∈𝜇 containing 𝑥 and an open set 𝑉 in 𝑌 containing 𝑦 such that 𝑓(𝑈)∩cl(𝑉)=Ø.

Theorem 36. If a function 𝑓∶(𝑋,𝜇)→(𝑌,𝜎) is faintly (𝜇,𝜎)-continuous and (𝑌,𝜎) is 𝑇₂, then 𝐺(𝑓) is faintly strong 𝜇-closed.

Proof. Let 𝑥∈𝑋, and let 𝑦=𝑓(𝑥). Then, (𝑥,𝑦)∈(𝑋×𝑌)⧵𝐺(𝑓). Then, by Lemma 32, there exist a 𝜇-open set 𝑈 in 𝑋 containing 𝑥 and a 𝜃-open set 𝑉 in 𝑌 containing 𝑦 such that 𝑓(𝑈)∩𝑉=Ø. Since 𝑉 is 𝜃-open, there exists an open set 𝑉₀ in 𝑌 containing 𝑦 such that 𝑦∈𝑉₀⫅cl(𝑉₀)⫅𝑉. Thus 𝑓(𝑈)∩cl(𝑉₀)=Ø. Thus, by Lemma 35, 𝐺(𝑓) is faintly strong 𝜇-closed.

Theorem 37. If a function 𝑓∶(𝑋,𝜇)→(𝑌,𝜎) is a surjective function with faintly strong 𝜇-closed graph 𝐺(𝑓), then (𝑌,𝜎) is 𝑇₂.

Proof. Let 𝑦₁ and 𝑦₂ be any two distinct points of 𝑌. Then, since 𝑓 is surjective, there exists 𝑥₁∈𝑋 such that 𝑓(𝑥₁)=𝑦₁; hence, (𝑥₁,𝑦₂)∈(𝑋×𝑌)⧵𝐺(𝑓). Since 𝐺(𝑓) is faintly strong 𝜇-closed, there exist a 𝜇-open set 𝑈 in 𝑋 containing 𝑥₁ and an open set 𝑉 of 𝑌 containing 𝑦₂ such that 𝑓(𝑈)∩cl(𝑉)=Ø. Therefore, we have 𝑦₁=𝑓(𝑥₁)∈𝑓(𝑈)⫅𝑌⧵cl(𝑉). Hence, there exists an open set 𝐻 of 𝑌 such that 𝑦₁∈𝐻 and 𝐻∩𝑉=Ø. Moreover, we have 𝑦₂∈𝑉, and 𝑉 is open in 𝑌. This shows that (𝑌,𝜎) is 𝑇₂.

Theorem 38. If a function 𝑓∶(𝑋,𝜇)→(𝑌,𝜎) has a faintly 𝜇-closed graph, then it has also a faintly strong 𝜇-closed graph.

Proof. Let 𝑥∈𝑋, and let 𝑦≠𝑓(𝑥). Then, (𝑥,𝑦)∈(𝑋×𝑌)⧵𝐺(𝑓). Then there exist 𝑈 in 𝜇 containing 𝑥 in 𝑋 and a 𝜃-open set 𝑉 in 𝑌 containing 𝑦 such that 𝑓(𝑈)∩𝑉=Ø. Since 𝑉 is 𝜃-open, there exists an open set 𝑉₀ such that 𝑉₀⫅cl(𝑉₀)⫅𝑉. So 𝑓(𝑈)∩cl(𝑉₀)=Ø. Thus, by Lemma 35, 𝑓 has a faintly strong 𝜇-closed graph.

The converse of the above theorem is not true in general as shown in Example 3 of [20].

Theorem 39. If 𝑓∶(𝑋,𝜇)→(𝑌,𝜎) has a faintly 𝜇-closed graph, then 𝑓(𝐾) is closed in (𝑌,𝜎) for each subset 𝐾 which is 𝜇-compact relative to (𝑋,𝜇).

Proof. Suppose that 𝑦∉𝑓(𝐾). Then, for each 𝑥∈𝐾, (𝑥,𝑦)∉𝐺(𝑓). Since 𝐺(𝑓) is faintly 𝜇-closed, by Lemma 32, there exist a 𝜇-open set 𝑈_𝑥 in 𝑋 containing 𝑥 and a 𝜃-open set 𝑉_𝑥 in 𝑌 containing 𝑦 such that 𝑓(𝑈_𝑥)∩𝑉_𝑥=Ø. Then, the family {𝑈_𝑥∶𝑥∈𝐾} is a cover of 𝐾 by 𝜇-open sets in 𝑋. So, there exists a finite subfamily 𝐾₀ of 𝐾 such that 𝐾⫅∪{𝑈_𝑥∶𝑥∈𝐾₀}. Set 𝑉=∩{𝑉_𝑥∶𝑥∈𝐾₀}. Then, 𝑉 is 𝜃-open (hence open) in 𝑌 containing 𝑦. Therefore, 𝑓(𝐾)∩𝑉⫅∪{𝑓(𝑈_𝑥)∶𝑥∈𝐾₀}∩𝑉⫅∪{𝑓(𝑈_𝑥)∩𝑉∶𝑥∈𝐾₀}=Ø. It then follows that 𝑦∉cl(𝑓(𝐾)). Thus, 𝑓(𝐾) is closed in 𝑌.

5. Conclusion

Similar types of faintly continuous functions can be defined from a topological space (𝑋,𝜏) to another topological space (𝑌,𝜎) from the definition of faintly (𝜇,𝜎)-continuous function by taking different GTs on 𝑋. In fact, different results on weak forms of faintly continuous functions can be derived from faintly (𝜇,𝜎)-continuous functions by replacing 𝜇 by the corresponding GTs on 𝑋 (see [20–29]).

Conflict of Interests

The author Bishwambhar Roy declares that the paper does not have any financial relation with any commercial identity that might lead to conflict of interests.

Acknowledgment

The author is thankful to the referee for his/her comments to improve the paper.