# δω-Continuity and some Results on δω-Closure Operator

Authors: Manjeet Singh; Asha Gupta; Kushal Singh
Journal: Journal of Mathematics
 (2022)

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## Abstract

Al-Jarrah et al. defined a new topological operator, namely,δω-closure operator, and proved that it lies between the δ-closure operator and the usual closure operator. Al-Ghour et al. defined θω-closure operator and discussed its properties. In this paper, it is proved that the δω-closure operator lies between the θω-closure operator and the usual closure operator. Also, sufficient conditions are given for the equivalence between the δω-closure operator and the θω-closure operator. Moreover, we define three new types of continuity, namely, δω-continuity, ω-δ-continuity, and almost δω-continuity, and discuss their properties. It is proved that the concepts of usual continuity and δω-continuity are independent of each other. In addition, the relationships between different types of continuity have been investigated. Further, some examples and counter examples are given.


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## Body

## 1. Introduction

Generating new topologies from old ones is a very interesting and useful concept in topology. LetG,τ be a topological space (abbreviated as ts) with no separation axioms assumed. The closure of a subset E of G is denoted by the symbol E¯, and the interior of a subset E is denoted by the symbol intE. A subset E is regular open (regular closed) if intE¯=EintE¯=E. A point z in G is called a δ-cluster point [1] of a subset E if for every open subset O containing z, intO¯∩E≠∅. The collection of all δ-cluster points of a subset E is called δ-closure of E and is denoted by the symbol ClδE. A subset E is called δ-closed [1] if ClδE=E and its complement is δ-open. The topology on G of all δ-open sets is denoted by the symbol τδ. The basis of τδ is the family of all regular open sets. A ts is called semiregular if τδ=τ. The δ-closure operator and its related concepts have been studied and generalized by many authors [2–6]. All regular open sets, with the exception of the empty set, are included in the class of somewhere dense sets. Somewhere dense sets have been studied by many authors [7, 8]. In 1982, Hdeib [9] classified ω-closed sets as well as the complement of an ω-closed set, i.e., an ω-open set. τω stands for the collection of all ω-open sets. The collection τω forms a topology finer than the usual topology τ. Further, the ω-open and ω-closed sets and their properties have been studied by many authors [10–12]. In paper [13], ω-regularity has been defined as a generalization of regularity by Al-Ghour. In 2019, Al-Jarrah et al. [14] defined δω-closure of a set by using ω-closure as follows: A point z in G is called δω-contact point of a subset E if the interior of every ω-closed neighborhood (abbreviated as nbhd) of z intersects E. The set of all δω-contact points of E is called the δω-closure of E and is denoted by the symbol ClδωE. The δω-closure operator is a new topological operator which strictly lies between the usual closure and the δ-closure operators. Al Ghour et al. [13, 15] have defined new topological operators, namely, θω-closure operator and θN-closure operator, and discussed their properties.This paper is organized as follows: After the introduction, definitions and results that have been already defined and proved are included in preliminaries and will be utilized to support various findings in the following sections. In the next section, we have proved thatδω-closure operator lies between the usual closure operator and θω-closure operator. Sufficient conditions are given for the equivalence of δω-closure operator and θω-closure operator, and also of δω-closure operator and the usual closure operator. In last section, we define some new types of continuity, namely, δω-continuity, ω-δ-continuity, and almost δω-continuity. We observed that continuity and δω-continuity are independent of each other. Moreover, we give some sufficient conditions for a δω-continuous function to be continuous. In addition, we establish the relationships between different types of continuity and discuss some of their properties. Some examples and counter-examples are also provided.


## 2. Preliminaries

Definition 1 (see [9]).
LetG,τ be a ts and E⊆G, then, the set of condensation points of E is denoted by the symbol CondE and defined as CondE=z∈G:O∩Eisuncountable,O∈τandz∈O.
The setE is called ω-closed if CondE⊆E and its complement is ω-open. τω represents the collection of all ω-open sets.Theorem 2 (see [9]).
LetG,τ be a ts and E⊆G, then
(a)
τω forms a topology on G(b)
τ⊆τω and τω≠τ in general
The closure ofE in G,τω is denoted by the symbol E¯ω. Also, E¯ω⊆E¯ and E¯ω≠E¯ in general.Definition 3 (see [1]).
LetG,τ be a ts and E⊆G, then, the set z∈G:O¯∩E≠∅,O∈τandz∈O is called the θ-closure of E and denoted by the symbol ClθE.
The setE is called θ-closed if ClθE=E and its complement is θ-open. τθ represents the collection of all θ-open sets.Theorem 4 (see [1]).
LetG,τ be a ts and E⊆G, then
(a)
τθ forms a topology on G(b)
τθ⊆τ and τθ≠τ in generalTheorem 5 (see [1]).
A ts is regular ifτθ=τ.Definition 6 (see [13]).
LetG,τ be a ts and E⊆G, then, the set z∈G:O¯ω∩E≠∅,O∈τandz∈O is called the θω-closure of E and denoted by the symbol ClθωE.
The setE is said to be θω-closed if ClθωE=E and its complement is θω-open. τθω represents the collection of all θω-open sets.Theorem 7 (see [13]).
LetG,τ be a ts and E⊆G, then
(a)
τθω forms a topology on G(b)
τθ⊆τθω⊆τ and τθω≠τ in generalTheorem 8 (see [13]).
A ts isω-regular if τθω=τ.Definition 9 (see [14]).
LetG,τ be a ts and E⊆G, then, the set z∈G:intO¯ω∩E≠∅,O∈τandz∈O is called the δω-closure of E and denoted by the symbol ClδωE.
A setE is called δω-closed if ClδωE=E and its complement is δω-open. τδω represents the collection of all δω-open sets.Theorem 10 (see [14]).
LetG,τ be a ts and let E⊆G. Then
(a)
E¯⊆ClδωE⊆ClδE(b)
IfE is δ-closed, then E is δω-closed(c)
IfE is δω-closed, then E is closedDefinition 11.
A tsG,τ is called
(a)
[16] Locally countable if for each z∈G, there exists an open subset O such that z∈O and O is countable(b)
[13] Antilocally countable if every nonempty open subset is uncountable(c)
[17] Locally indiscrete if every open subset is closedLemma 12 (see [13]).
(a)
IfG,τ is an antilocally countable ts, then for all E∈τω, E¯ω=E¯(b)
IfG,τ is a locally countable ts, then τω is a discrete topologyDefinition 13 (see [13]).
A ts is calledω-locally indiscrete if every open subset is ω-closed.Theorem 14 (see [13]).
(a)
Every ts that is locally indiscrete isω-locally indiscrete(b)
Every ts that is locally countable isω-locally indiscreteTheorem 15 (see [14]).
LetG,τ be a ts and E⊆G. Then, E∈τδω if for each z∈E, and there is O∈τ such that z∈O⊆intO¯ω⊆E.Theorem 16 (see[13]).
LetG,τ be an ω-locally indiscrete ts and let E⊆G. Then
(a)
E¯=ClθωE(b)
IfE is closed in G,τ, then, E is θω-closed in G,τTheorem 17 (see [13]).
For any tsG,τ, the following are equivalent:
(a)
G,τ is ω-regular(b)
For each subsetE, E¯=ClθωELemma 18 (see [1]).
LetG,τ be a ts. Then, for each E∈τ, E¯=ClθE.Theorem 19 (see [13]).
LetG,τ be a ts. Then, for each E∈τω, E¯=ClθωE.Theorem 20 (see [13]).
(a)
Every openω-closed subset in a ts is θω-open(b)
Every countable open subset in a ts isθω-openDefinition 21 (see [18]).
A functionf:G⟶H is called δ-continuous if for each z∈G and each open nbhd P of fz, there is an open nbhd O of z such that fintO¯⊆intP¯.Definition 22 (see [13]).
A tsG,τ is ω-regular if for each open nbhd O of z there exists an open subset P such that z∈P⊆P¯ω⊆O.Definition 23 (see [19]).
A functionf:G⟶H is called θω-continuous if for each z∈G and each open nbhd P of fz, there is an open nbhd O of z such that fO¯⊆P.¯ωDefinition 24 (see [20]).
A functionf:G⟶H is called almost open if for every regular open subset O of G, fO is open in H.Definition 25 (see [20]).
LetG,τ and H,σ be two topological spaces (abbreviated as ts′s), and a function f:G⟶H is called almost continuous if for each z∈G and for each P⊆H open nbhd of fz, there exists an open nbhd O of z such that fO⊆intP¯.


## 3. Some Results onδω-Closure Operator

In this section, the relationships between usual closure operator,δω-closure operator, and θω-closure operator have been discussed.Theorem 26.
LetG,τ be a ts and let E⊆G. Then
(a)
ClδωE⊆ClθωE(b)
IfE is θω-closed, then E is δω-closedProof.
(a)
To proveClδωE⊆ClθωE, let z∈ClδωE and O∈τ with z∈O. Therefore, intO¯ω∩E≠∅. We know intO¯ω⊆O¯ω, and then, O¯ω∩E≠∅. We have z∈ClθωE(b)
LetE be θω-closed, then, ClθωE=E. By using part (a) and Theorem 10(a), ClδωE=E, and hence, E is δω-closed.
The following results give the equivalence conditions for theδω-closure operator with δ-closure and θω-closure operators.Theorem 27.
LetG,τ be a ts and E⊆G. If G,τ is ω-locally indiscrete, then
(a)
E¯=ClδωE=ClθωE(b)
E is closed in G,τ if E is δω-closed in G,τ(c)
E is δω-closed in G,τ if E is θω-closed in G,τProof.
(a)
By Theorems26, 10(a) and 16(b)
SupposeE is closed, then, E=E¯. By part (a), E=ClδωE, and therefore, E is δω-closed. The converse follows from Theorem 10(c).(c)
SupposeE is δω-closed, then E=ClδωE. By part (a), E=ClθωE, and therefore, E is θω-closed. The converse follows from Theorem 26(b).Corollary 28.
LetG,τ be a ts and E⊆G. If G,τ is ω-regular, then
(a)
E¯=ClδωE=ClθωE(b)
E is closed in G,τ if E is δω-closed in G,τ(c)
E is δω-closed in G,τ if E is θω-closed in G,τRemark 29.
The statements of Theorem27 hold even if G,τ is locally indiscrete or locally countable by using Theorem 14.Theorem 30.
LetG,τ be a ts and E⊆G. If G,τ is antilocally countable, then
(a)
ClδE=ClδωE(b)
E is δω-closed in G,τ if E is δ-closed in G,τProof.
(a)
By Theorem10(a), we have ClδωE⊆ClδE. Now to prove ClδE⊆ClδωE, let z∈ClδE and open subset O such that z∈O. By definition, intO¯∩E≠∅. Given G,τ is antilocally countable, then by Lemma 12(a), O¯ω=O¯, and hence, intO¯ω∩E≠∅. Therefore, we get z∈ClδωE(b)
SupposeE is δω-closed, then E=ClδωE. By part (a), E=ClδE. Therefore, E is δ-closed. The converse follows from Theorem 10(b).
The following result gives some properties ofδω-closure operator.Theorem 31.
LetG,τ be a ts and let E,F⊆G. Then
(a)
IfE⊆F⊆G, then ClδωE⊆ClδωF(b)
For everyE,F⊆G, ClδωE∪F=ClδωE∪ClδωF(c)
For everyE⊆G, ClδωE is a closed subset in G,τ(d)
For everyE∈τω, ClδωE=E¯(e)
For everyE∈τ, ClδE=ClδωE=ClθωE=E¯Proof.
(a)
Letz∈ClδωE and O∈τ with z∈O. Since z∈ClδωE, then intO¯ω∩E≠∅. Since E⊆F, therefore, intO¯ω∩F≠∅. Hence, z∈ClδωF(b)
By part (a), we get ClδωE∪F⊇ClδωE∪ClδωF. Now to prove ClδωE∪F⊆ClδωE∪ClδωF. Let z∉ClδωE∪ClδωF, and there are two open sets O,P containing z such that intO¯ω∩E=∅ and intP¯ω∩F=∅. Now, we have z∈O∩P∈τ and(1)intO∩P¯ω∩E∪F=intO∩P¯ω∩E∪intO∩P¯ω∩F⊆intO¯ω∩E∪intP¯ω∩F=∅∪∅=∅.
Hence, we getz∉ClδωE∪F.
(c)
To proveG−ClδωE∈τ. Let z∈G−ClδωE, there is an open set O containing z such that intO¯ω∩E=∅. Therefore, O∩ClδωE=∅. Hence, we get G−ClδωE∈τ(d)
By Theorems19, 10, and 26(e)
Follows from part (d) and Lemma 18Theorem 32 (see [14]).
LetG,τ be a ts, then, τδ⊆τδω⊆τ.
The equality in Theorem32 does not hold, as the following examples show.Example 1.
LetR,τ be a ts where R is real line and τ=∅∪O⊆R:complementofOiscountable. The regular open subsets of ts τ are ∅,R, and then, τδ=∅,R. Also R,τ is antilocally countable ts. Therefore, by Theorem 30, we have τδ=τδω. Then, τδω=∅,R. Hence, we get τδω≠τ.Example 2.
LetZ,τ be a ts where Z is set of integer and τ=∅∪O⊆Z:complementofOisfinite. The regular open subsets of ts τ are ∅,Z, and then, τδ=∅,Z. Also Z,τ is locally countable ts. Therefore, by Remark 29, we have τ=τδω. Hence, τδ≠τδω.
If tsG,τ is semiregular, then τδ=τδω=τ.Theorem 33.
LetG,τ be a ts, then, τθω⊆τδω⊆τ.Proof.
To proveτθω⊆τδω. Let E∈τθω, then, G−E is θω-closed and by Theorem 26b, G−E is δω-closed. Hence, E∈τδω.
The equality in Theorem33 does not hold, as the following example shows.Example 3.
LetR,τ be a ts where R is the real line and τ=∅,R,N,Qc,N∪Qc. By Example 2.26 of [13], τθω=∅,R,N. Now, to prove Qc∈τδω by using Theorem 15, we have to find O∈τ for each z∈Qc such that z∈O⊆intO¯ω⊆Qc. Let O=Qc, then, Qc¯ω=R−N. Now, intQc¯ω=intR−N=Qc. So, z∈O⊆intO¯ω⊆Qc is true for O=Qc, and this implies Qc∈τδω. But Qc∉τθω. Hence, τδω≠τθω.
If tsG,τ is ω-regular (ω-locally indiscrete or locally indiscrete or locally countable), then τθω=τδω=τ. If ts G,τ is antilocally countable, then τδ=τδω. If ts G,τ is regular, then τθ=τδ=τθω=τδω=τ. If a ts is antilocally countable and ω-regular, then τθ=τδ=τθω=τδω=τ. It can be easily seen by Example 2 and Example 3 that τδ and τθω are incomparable.Theorem 34.
(a)
Every openω-closed subset in a ts is δω-open(b)
Every countable open subset in a ts isδω-openProof.
Directly follow from Theorems33 and 20.


## 4.δω-Continuity

In this section, we define some new types of continuity and discuss their relationships.Definition 35.
A functionf:G⟶H is called δω-continuous if for each z∈G and each open subset P of H containing fz, there is an open subset O of G containing z such that fintO¯⊆intP¯ω.Theorem 36.
Everyδω-continuous function is δ-continuous.Proof.
LetG,τ and H,σ be two ts′s and f:G⟶H be a δω-continuous function. Let z be in G and let P⊆H open nbhd of fz. By assumption, there exists O⊆G an open nbhd of z such that fintO¯⊆intP¯ω⊆intP¯. This implies that f is δ-continuous.
The converse of the Theorem36 does not hold, as the following example shows.Example 4.
Let a functionf:Z,τind⟶Z,τcof (Z be the set of integers) be defined as fz=z, where τind is indiscrete topology on Z and τcof is finite complement topology on Z. Then, f is δ-continuous but not δω-continuous. Let z∈Z and P∈τcof such that fz=z∈P. Take O=Z such that z∈O∈τcof and fintO¯=Z⊆intP¯=Z. Hence, f is δ-continuous. Now to prove f is not δω-continuous, let z=0 and P=Z−1,2. Then, P∈τcof with f0=0∈P. Suppose there exists O∈τind such that 0∈O∈τind and fintO¯⊆intP¯ω, but open subset in τind containing 0 is Z. Take O=Z, then fintO¯⊆intP¯ω is not true because fintO¯=Z and intP¯ω=Z−1,2. This implies that f is not δω-continuous.The following result gives sufficient criteria for aδ-continuous to be δω-continuous.Theorem 37.
LetG,τ and H,σ be two ts′s, and let f:G⟶H be a δ-continuous function with H,σ antilocally countable, then, f is δω-continuous.Proof.
Letz be in G and let P⊆H be any open nbhd of fz. By assumption, f is δ-continuous, so there exists O⊆G an open nbhd of z such that fintO¯⊆intP¯. Given H,σ is antilocally countable, then by Lemma 12, we have P¯=P¯ω, and thus, fintO¯⊆intP¯ω. This implies that f is δω-continuous.
The independence of continuity andδ-continuity have been observed by Noiri [18]. In the following Examples 5 and 6, it is shown that the concepts of continuity and δω-continuity are independent of each other.Example 5.
LetG be the real line with the usual topology and H the real line with the co-countable topology. Let f:G⟶H be the function defined by fz=z. The ts H is antilocally countable and by Example 4.4 of [18], f is δ-continuous but not continuous, and then, by Theorem 37, f is δω-continuous.Example 6.
LetN be the set of natural number with topology ∅,N,1, and let a function f:N⟶N be defined as fz=z. It can be easily seen that f is a continuous function. To check that f is δω-continuous, take z=1 and P=1, and then, P is open and fz=z∈P. Also N is locally countable, and then, P¯ω=P. Now for an open subset O with 1∈O, we have O=1 or O=N. In both possibilities, O¯=N and fintO¯=N⊆intP¯ω=1 which is not possible. This implies that f is not δω-continuous.The following results from38 to 41 give sufficient criteria for a δω-continuous function to be continuous.Theorem 38.
LetG,τ and H,σ be two ts′s and let f:G⟶H be a δω-continuous function with H,σ locally countable, then, f is continuous.Proof.
Letz be in G and let P⊆H be an open nbhd of fz. By assumption, f is δω-continuous, so there exists O⊆G open nbhd of z such that fintO¯⊆intP¯ω. Since H,σ is locally countable, then by Lemma 12(b), we have τω is the discrete topology. Therefore, P=P¯ω, and thus, fO⊆fintO¯⊆intP¯ω⊆P. This implies that f is continuous.Theorem 39.
LetG,τ and H,σ be two ts’s and let f:G⟶H be a δω-continuous function with H,σω-locally indiscrete, then f is continuous.Proof.
Letz be in G and let P⊆H be an open nbhd of fz. By assumption, f is δω-continuous, so there exists O⊆G an open nbhd of z such that fintO¯⊆intP¯ω. Since H,σ is ω-locally indiscrete, then P is ω-closed and P¯ω=P. Thus fO⊆fintO¯⊆intP¯ω⊆P. Hence f is continuous.Corollary 40.
LetG,τ and H,σ be two ts’s and let f:G⟶H be a δω-continuous function with H,σ locally indiscrete, then, f is continuous.Theorem 41.
LetG,τ and H,σ be two ts’s and let f:G⟶H be a δω-continuous function with H,σω-regular, then f is continuous.Proof.
Letz be in G and let P⊆H be an open nbhd of fz. By assumption, H,σ is an ω-regular ts, then by Definition 22, there exists N of H open subset such that fz∈N⊆N¯ω⊆P. Given f is δω-continuous, there exists O⊆G an open subset containing z such that fintO¯⊆intN¯ω. Thus, we have fO⊆fintO¯⊆intN¯ω⊆P. This implies that f is continuous.
The next result gives sufficient criteria for aθω-continuous function to be δω-continuous.Theorem 42.
Iff:G,τ⟶H,σ is a θω-continuous function and almost-open, then f is δω-continuous.Proof.
Letz be in G and let P⊆H be an open nbhd of fz. By assumption, f is θω-continuous, so there exists O⊆G an open nbhd of z such that fO¯⊆P¯ω. Therefore, fintO¯⊆fO¯⊆P¯ω. Since f is almost-open, then intfintO¯=fintO¯ and thus fintO¯⊆intP¯ω. Hence, f is δω-continuous.Corollary 43.
Iff:G,τ⟶H,σ is a θω-continuous function and open, then f is δω-continuous.Definition 44.
LetG,τ and H,σ be two ts’s and a function f:G⟶H is called ω-δ-continuous function if for each z∈G and for each P⊆H open nbhd of fz, there is an open subset O of G containing z such that fintO¯ω⊆intP¯.Theorem 45.
Everyω-δ-continuous function is almost continuous.Proof.
LetG,τ and H,σ be two ts’s and let f:G⟶H be an ω-δ-continuous function. Let z be in G and let P⊆H be an open nbhd of fz. By assumption, f is ω-δ-continuous, so there is an open nbhd O in G containing z such that fO⊆fintO¯ω⊆intP¯. Hence, f is almost continuous.
The following results give sufficient criteria for the almost continuous function to beω-δ-continuous.Theorem 46.
LetG,τ and H,σ be two ts’s and let f:G⟶H be almost continuous with G,τω-locally indiscrete, then f is ω-δ-continuous.Proof.
Letz be in G and let P⊆H be an open nbhd of fz. By assumption, f is almost continuous, so there is an open nbhd O⊆G of z such that fO⊆intP¯. Since G,τ is ω-locally indiscrete, then O is ω-closed and O¯ω=O. Thus, fintO¯ω=fO⊆intP¯. Hence, f is ω-δ-continuous.
The following corollaries can be easily proved by Theorems14 and 46.Corollary 47.
LetG,τ and H,σ be two ts’s and let f:G⟶H be almost continuous with G,τ locally indiscrete, then f is ω-δ-continuous.Corollary 48.
LetG,τ and H,σ be two ts’s and let f:G⟶H be almost continuous with G,τ locally countable, then f is ω-δ-continuous.Theorem 49.
LetG,τ and H,σ be two ts’s and let f:G⟶H be almost continuous with G,τω-regular, then f is ω-δ-continuous.Proof.
Letz be in G and let P⊆H be an open nbhd of fz. By assumption, f is almost continuous, so there is an open nbhd N⊆G of z such that fN⊆intP¯. Since G,τ is ω-regular, then there is an open subset O in G containing z such that intO¯ω⊆O¯ω⊆N. Therefore, fintO¯ω⊆fN⊆intP¯. Hence, f is ω-δ-continuous.Theorem 50.
Everyδ-continuous function is ω-δ-continuous.Proof.
Letz be in G and let P⊆H be an open nbhd of fz. By assumption, f is δ-continuous, so there is an open nbhd O⊆G of z such that fintO¯⊆intP¯. Thus, fintO¯ω⊆fintO¯⊆intP¯. Hence, f is ω-δ-continuous.
The converse of the Theorem50 does not hold, as the following example shows.Example 7.
The functionf defined in Example 4.5 of [18], shows that f is continuous but not δ-continuous, and then, f is almost continuous by definition. Also, G is locally countable, so by Corollary 48, f is ω-δ-continuous.The following result gives sufficient criteria for aω-δ-continuous function to be δ-continuous.Theorem 51.
LetG,τ and H,σ be two ts’s and let f:G⟶H be ω-δ-continuous with G,τ antilocally countable, then f is δ-continuous.Proof.
Letz be in G and let P⊆H be an open nbhd of fz. By assumption, f is ω-δ-continuous, so there is an open nbhd O of z such that fintO¯ω⊆intP¯. Since G,τ is antilocally countable, then by Lemma 12, O¯ω=O¯. Therefore, fintO¯=fintO¯ω⊆intP¯. Hence, f is δ-continuous.Definition 52.
A functionf:G⟶H is called almost δω-continuous if for each z∈G and each open subset P of H containing fz, there is an open subset O of G containing z such that fO⊆intP¯ω.Theorem 53.
Every almostδω-continuous function is almost continuous.Proof.
Letz be in G and let P⊆H be an open nbhd of fz. By assumption, f is almost δω-continuous, so there is an open nbhd O of z such that fO⊆intP¯ω. By Theorem 2, P¯ω⊆P¯, and therefore, fO⊆intP¯ω⊆intP¯. Hence, f is almost continuous.
The converse of the Theorem53 does not hold, as the following example shows.Example 8.
Consider a functionf:N,τ⟶N,τcof defined by fz=z, where τ is indiscrete topology and τcof is finite complement topology on N. To check f is almost continuous, let z∈N and P∈τcof such that fz=z∈P. Then, for every P∈τcof, P¯=N. Choose O=N, then z∈O∈τ and fO=fN=N⊆intP¯=intN=N. Now, to check f is not almost δω-continuous, let z=1 and P=N−2. Since τcof is locally countable, then, P¯ω=P. So O∈τ containing z is N, then, fO=fN=N⊆intP¯ω=intP=intN−2=N−2 which is not true. Hence, f is not almost δω-continuous.The following result gives sufficient criteria for the almost continuous function to be almostδω-continuous:Theorem 54.
LetG,τ and H,σ be two ts’s and let f:G⟶H be almost continuous with H,σ antilocally countable, then f is almost δω-continuous.Proof.
Letz be in G and let P⊆H be an open nbhd of fz. By assumption, f is almost continuous, so there is an open nbhd O of z such that fO⊆intP¯. Since H,σ is antilocally countable, then P¯ω=P¯. Therefore, fO⊆intP¯=intP¯ω. Hence, f is almost δω-continuous.Theorem 55.
Every continuous function is almostδω-continuous.Proof.
Letz be in G and let P⊆H be an open nbhd of fz. By assumption, f is continuous, so there is an open nbhd O of z such that fO⊆P. Thus, fO⊆P⊆intP¯ω. Hence, f is almost δω-continuous.
The converse of the Theorem55 does not hold, as the following example shows.Example 9.
Consider a functionf:R,τ⟶R,τcoc defined by fz=z, where τ is the usual topology and τcoc is the countable complement topology on R. It is easy to check that f is discontinuous. To check, f is almost δω-continuous. Let z∈R and P∈τcoc such that fz=z∈P. Since R,τcoc is antilocally countable, then, P¯ω=P¯. Therefore, for any open subset, P∈τcoc such that P¯ω=P¯=R. Take O=R such that z∈O∈τ and fO=R⊆intP¯ω=intR=R. Hence, f is almost δω-continuous.The following result gives sufficient criteria for the almostδω-continuous function to be continuous.Theorem 56.
LetG,τ and H,σ be two ts’s and let f:G⟶H be almost δω-continuous with H,σω-locally indiscrete, then f is continuous.Proof.
Letz be in G and let P⊆H be an open nbhd of fz. By assumption, f is almost δω-continuous, so there is an open nbhd O of z such that fO⊆intP¯ω. Since H,σ is ω-locally indiscrete, then P¯ω=P. Therefore, fO⊆intP¯ω=P. Hence, f is continuous.
The following corollaries can be easily proved by Theorems14 and 56.Corollary 57.
LetG,τ and H,σ be two ts’s and let f:G⟶H be almost δω-continuous with H,σ locally indiscrete, then f is continuous.Corollary 58.
LetG,τ and H,σ be two ts’s and let f:G⟶H be almost δω-continuous with H,σ locally countable, then f is continuous.Theorem 59.
Everyδω-continuous function is almost δω-continuous.Proof.
Letz be in G and let P⊆H be an open nbhd of fz. By assumption, f is δω-continuous, so there is an open nbhd O in G containing z such that fintO¯⊆intP¯ω. Thus, fO⊆fintO¯⊆intP¯ω. Hence, f is almost δω-continuous.
The converse of the Theorem59 does not hold, as the following example shows the same.Example 10.
In Example6, f is continuous but not δω-continuous. Also, by Theorem 55, every continuous function is almost δω-continuous.


## 5. Conclusion

In this work, we continue the research onδω-closure operator, and some properties of δω-closure operator are discussed. Also, we give some sufficient conditions for the equivalence of δω-closure and θω-closure operators. Further, we define some new types of continuity, namely, δω-continuity, ω-δ-continuity, and almost δω-continuity. Also, some examples and counter examples are given.



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