# α-Completely Regular and Almost α-Completely Regular Spaces

Authors: A.A. Azzam; A.A. Nasef
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This work is aimed at studying some of the characterizations ofα-completely regular and almost α-completely regular spaces through the new concept of α-zero sets and α-continuous functions. 2010 Mathematics Subject Classification: primary 05C38, 15A15; Secondary 05A15, 15A18.


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## Body

## 1. Introduction and Preliminaries

In the property, zero sets and co-zero sets were used to characterize the completely regular spaces. In topology, Malghan et al. [1] defined and investigated the concepts of semizero sets and co-semizero sets to use semicontinuous functions to characterize the features of s-completely regular spaces and virtually s-completely regular spaces. A subset D of a space Y which carries topology ζ is named zero-open if for every y∈D, there is a set of zero Z (in Y) and a co-zero set C (in Y) that is how that y∈C⊆Z⊆D. Zero-closed sets are complementary to zero-open sets, and the family Z of each open zero sets in the topology ζ defined on Y generates a coarser topology denoted by ζz [2]. Our aims in this work are to define and study the α-zero sets and co-α-zero sets in the study of characterizations of α-completely regular spaces and almost α-completely regular spaces.IfY,ζ be a topological space unless specifically indicated, then no separations are assumed. Let D be a subset of Y,ζ topological space. Let C−D and I−D represent a subset’s closure and interior D.Definition 1.
TheD subset of the space Y,ζ is named as follows:(i)
semiopen [3] if D⊑C−I−D(ii)
preopen [4] if D⊑I−C−D(iii)
regular open [5] if D=I−C−D(iv)
regular closed [6] if D=C−I−D(v)
α open [7] if D⊑I−C−I−DRemark 1.
Add-onα-open is named α-closed. αOY (resp. SO(Y)) denotes the entire of α-open (resp. semiopen) sets of Y,ζ.Definition 2.
Ifξ−1V is α-open in Y for each open set V of X, the function ξ:Y⟶X is called α-continuous [4].Definition 3.
The topological spaceY,ζ is referred to as follows:(i)
α-regular [8] if for each closed set E, and each point y does not belong to E, there is disjoint preopen sets U and V, and as a result, Y∈U and E⊑V.(ii)
α-completely regular [1] (resp. completely regular [9]) if for every closed set E, and each point y∈Y/E, there is a precontinuous (resp. semicontinuous) function ξ:Y⟶0,1 that is how that ξy=0 and ξx=1 for each x∈E.(iii)
Almostα-completely regular [1] (resp. almost s-completely regular [9]) if for each regular closed set E, and each point y∈Y/E, there exists an α-continuous (resp. semicontinuous) function ξ:Y⟶0,1 such that ξy=0 and ξx=1 for each x∈E.Definition 4.
If any dense subset of a spaceY with topology ζ is open, it is called submaximal. If the closure of each opening set is open in a space Y, it is called extremely disconnected (E. D.) space. If there is a semicontinuous function ξ:Y⟶R such that D=y∈Y:ξy=0, a subset D of a space Y is called semizero [1] of Y.


## 2.α-Zero Sets

In this section, we introduceα-zero sets and look into some of its properties.Definition 5.
The subsetD of the Y space is alleged to be α-zero set of Y if there is α-continuous function ξ:Y⟶R that is how that D=y∈Y:ξy=0.Remark 2.
A subsetD of a space Y is called a co-α-zero set of Y if its complement is an α-zero set. Every zero set in Y is an α-zero set to Y if its complement is an α-zero set.Remark 3.
In a spaceY, if ξ:Y⟶R is an α-continuous function, then the set y∈Y:ξy=0 is an α-zero set.Remark 4.
Ifξ:Y⟶R is α-continuous function that may be denoted by αZξ. Thus, we write αZξ=y∈Y:ξy=0. Thus, αZξ is an α-zero set of Y. Therefore, it is clear that if B is an α-zero set in Y, then it can be expressed as B=αZξ, where ξ is α-continuous function.Lemma 1.
IfY is a submaximal space, then a mapping ξ:Y⟶X is α-continuous; then, each member of a basis for X has its inverse image α-open set to Y.Lemma 2.
IfY is a submaximal space, then a mapping ξ:Y⟶R is α-continuous if and only if for each b∈R both the sets ξ−1−∞,b and ξ−1b,∞ are α-open sets.Lemma 3.
The following are similar ifY is a submaximal space:(i)
ξ:Y⟶R is α continuous(ii)
For eachb∈R, ξ−1−∞,b and −ξ−1−∞,b are α open in Y(iii)
For eachb∈R, ξ−1b,∞ and −ξ−1−∞,b are α open in YLemma 4.
We consider the submaximal spaceY. If ξ,η:Y⟶R are α-continuous then:(i)
ξα is α continuous for each α≥0(ii)
aξ+bη is α continuous for each pair of reals b and a(iii)
ξ.η is α− continuous(iv)
1/ξ is α continuous whenever ξ≠0 on Y
These results can be proved by using the proofs of Lemmas1–3. (see[4], p.84]).Lemma 5.
IfY is a submaximal space and if ξi:Y⟶Ri=1k is a finite family of α-continuous functions, then the functions M,m:X⟶R defined by Mx= max ξixi=1k and Mx= min ξixi=1k are also α-continuous.Proof.
We note thatM−1b,∞=∪i=1kξix and M−1−∞,b=∩i=1kξix. Since ξi is α-continuous and Y is an submaximal space, M−1−∞,b, and M−1b,∞ are α-open sets; hence, M is an α-continuous.Lemma 6.
For real valued functions in a submaximal spaceY, the following propositions are true:(i
IfB is an α-zero set in Y, then there is the α-continuous function η:Y⟶R that shows that ηy≥0 for each y∈Y, and B=αZξ(ii)
IfB is an α-zero set in Y, then there is an α-continuous ρ:Y⟶0,1 function such that B=αZρ(iiiii)
Theα-zero sets in Y are the finite union of α-zero set in Y(iv)
Finite intersection ofα zero is an α-zero set in Y(v)
Ifa∈R and ξ:Y⟶R is an α-continuous function, then the sets B=y∈Y:ξy≥a and D=y∈Y:ξy≤a are α-zero sets in Y(vi)
Ifa∈R and ξ:X⟶R is an α-continuous function, then the sets B=y∈Y:ξy<a and D=y∈Y:ξy>a are co-α-zero sets in Y
These results can be proved by using Lemma4 and 5 (see [10], p.18).Theorem 1.
IfB and D are disjoint α-zero sets of an submaximal space Y, there is a disjoint co-α-zero sets U and V as well as B⊑U and D⊑V.Theorem 2.
In a submaximal spaceY, every α-zero (resp.co-α-zero) set is an α-closed (resp. α-open) set.Proof.
IfD is an α-zero set to Y, then by Lemma 6, we have D=αZη, where η:Y⟶R is α continuous, and ηy≥0 for all y∈Y. Then, ηy=0 for all y∈D. Hence, η−10=d. Since 0 is closed in R, and η is α continuous, consequently, D is α-closed set in Y. The second part is proved similarly.


## 3. Characterizations ofα-Completely Regular Spaces

We start three section with some characterizations of.ZDefinition 6.
(12). We assume thatB is a subset of the Y space. If there is an α-open set U of Y that B⊑U⊑V, then the V subset of the Y space is named α-neighbourhood of B.
IfB=y for some y∈Y, then V in the Definition 6 is the neighbourhood of the point y.Theorem 3.
In the submaximalα-completely regular space Y, every neighbourhood of a point contains an α-zero set α-neighbourhood of the point.Definition 7.
A familyζ of a space’s subsets is a net for Y if each open set is the union of a family of ζ elements.Theorem 4.
For a submaximal spaceY, the subsequents are interchangeable as follows:(i)
Y is α-completely regular space(ii)
Every closed setB of Y is the intersection of α-zero sets which are α-neighbourhoods of B(iii)
All inclusive family of co-α-zero sets of Y is a net for the space YProof.
The proof is obvious.


## 4. Characterizations of Almostα-Completely Regular Spaces

In this section, we are introducing to characterize the almostα-completely regular spaces using the α-zero sets and co-α-zero sets in the following theorem.We recall that a pointy∈Y is called a δ-accumulation point of a subset B of Y [11] at B⊓U≠ϕ for every open set U⊒y. The δ closure of B is indicated by CδB and consists of all δ-accumulation points of B. The set B is named δ closed [11, 12] if B=CδB.The family of all open sets onY is a topology, coarser than ζ denoted by ζδ.Complements ofδ-closed sets are called δ open, and the family of all δ-open sets on Y is a topology.Theorem 5.
For a submaximal spaceY, the subsequent are equal:(i)
Y is almost α-completely regular space(ii)
Everyδ-closed subset B of Y is expressible as the intersection of some α-zero sets which are α-neighbourhood of B(iii)
Everyδ-closed B subset of Y is the point at which everything comes together α-zero sets that are α-neighbourhoods of B(iv)
There is a co-α-zero set containing every δ-open subset of Y that contains a pointProof.
It is clear.


## 5.α-US-Spaces and Results

Definition 8.
A sequenceyn is called α-converges to a point y of Y, written as <yn>⟶αy if yn appears in each α-open set containing y at some point.
Any sequence<yn>α-converges to a y point; therefore, yn converges to y is obvious.Definition 9.
TheY space is named α-US if each sequence yn in Yis the value of α-converges to a single point.
We remember that spaceY is known as US-space [13] if every convergent sequence converges to exactly one limit point.Corollary 1.
Everyα-US-Space is US-Space and vice versa is not always true.Example 1.
IfR is the real set and ζ is the rational sequence topology on R (see [14], example 65), then R,ζ is α-US with each convergent sequence having exactly one limit point. However, we can see that R,ζ does not α-converge to a single point. This demonstrates that not every US space is alpha-US.Theorem 6.
Everyα-US-Space is α−T1.Proof.
We allowY to be α-US Space. Let y and x be two separate points of Y. We consider the sequence yn, where yn=y for every n. Clearly, <yn>α-converges to y. Also, since y≠x and Y is α-US, yn cannot α converge to x; that is, there exists an α-open set V containing x but not y. Similarly, we get a α-open set U containing y but not x if we consider the sequence xn where xn=x for every n and proceed as previously. As a result, the Y space is α−T1.Theorem 7.
Everyα−T2 space is α-US.Proof.
LetY denote α−T2-space and yn denote a Y sequence. We assume that <yn>α-converges to two separate points y and x if possible. That is, <yn> appears in every α-open set containing y, as well as every α-open set that contains x. Because Y is α−T2 space, that is a contradiction. As a result Y is α-US.Definition 10.
If each sequence inEα-converges to a point in E, the set E is sequentially α closed.Theorem 8.
The spaceY is α-US if and only if a successively α-closed subset of Y×Y is the diagonal set.Proof.
LetY be α-US and yn,yn be a Δ sequence. The sequence Y is then yn. Because Y is α-US, <yn>⟶αy for a unique y∈Y, implying that <yn>α-converges to y and x. As a result, x=y. As a result, Δ is an α-closed set.
We allowΔ to be α-closed consecutively and allow <yn>α-converging to y and x by allowing the sequence yn to converges to y and x. As a result, the sequence <yn,yn>α-converges to y,x. Because Δ is α-closed in sequential manner, y,x∈Δ implies that x=y implying that Y is α-US.Definition 11.
TheG subset of the Y space is named sequentially α-compact if each sequence in G contains a subsequence that α-converges to a point in G.Theorem 9.
Each sequentiallyα-compact set in α-US-space is sequentially α-closed.Proof.
LetY represent α-US-space and X represent a sequentially α-compact subset of Y. Let yn be a X sequence. We assume that <yn>α-converges to a point in Y−X. Because X is sequentially α-compact, let ynp be a subsequence of yn that α-converges to a point x∈X. In addition, we suppose a subsequence of ynp of <yn>α-converging to a point y in Y−X, x=y because xnp is a sequence in the α− US space Y. As a result, X is a sequentially α− closed set.
After that, we provideα-US space hereditary properties.Theorem 10.
Anα-US-α subset spaces is α-US.Proof.
LetX⊑Y be a α-set and Y be α-US-space. Let yn be a sequence in X. Let us pretend that <yn>α-converges to y,x in X. We will show that <yn>α-converges to y, and x with Y. We allow U to be any α-open subset of Y that contains y, and V to be any α-open set of Y that contains x. Then, according to Lemma 2.16 [4], U⊓X and V⊓X are α-open sets in X. As a result, yn, as well as U and V, are finally in U⊓X and V⊓X. This means x=y because Y is α-US. As a result, the X subspace is α-US.Theorem 11.
TheY topological space is α−T2 if and only if it is both α−R1 and α-US.Proof.
LetY be α−T2 of a space. Hence, Y is α−R1 by lemma in [7] and α-US by Theorem 5.4. Conversely, let Y be both α−R1 and α-US space. Then, to prove that Y is α−T2, we have by Theorem 5.3 that above every α-US space is α−T1, and Y is both α−T1 and α−R1, and it follows from lemma in [7] that space Y is α−T2.Definition 12.
If and only ifyn is common in every α-open set containing y, the x point is the α-cluster point of sequence yn. α−clyn denotes the collection of all α-cluster points of yn.Definition 13.
Theα-point in the sequence yn is the x point, if x is a <yn>α-cluster point. However, there is no yn subsequence, the value of α converges to x.Definition 14.
Each sequence that<yn>α-converges with a subsequence of <yn>α-side points and the Y topological space is said to be α−S1 if it is α-US.Definition 15.
TheY topological space is said to be α−S2 if it is α-US, and every sequence yn in Yα-converges which has no α-side points.
Obviously, by definition it follows that the results:Lemma 7.
(i)
Everyα−S2 space is α−S1(ii)
Everyα−S1 space is α-US
In the subsequent, we have defined sequentiallyα-continuous functions based on the concept of sequentially continuous functions.Definition 16.
Ifξynα-converges to ξy whenever ξn, a function ξ:Y⟶X is said to be sequentially α-continuous at y∈Y that is α-converging to y sequence. ξ is said to be sequentially α-continuous if y∈Y is sequentially α-continuous at all.Theorem 12.
Letξ:Y⟶X and η:Y⟶X be two sequentially α-continuous functions. If X is α-US, then the set B=y:ξy=ηy is sequentially α-closed.Proof.
We assumeX is α-US, and there is a sequence yn in B that is α-converging to y∈Y. Because ξ and η are sequentially α-continuous functions in the same order, then ξyn⟶aξy and ηyn⟶αηy. Thus, ξy=ηy and y∈B. As a result, B is sequentially α-closed.
Now, we are going to show thatα-US spaces have a product theorem.Theorem 13.
α-US is the product of the arbitrary family of α-US spaces.Proof.
LetY=Πλ∈∧Yλ, where Yλ is α-US. Let the sequence yn in Yα-converges to y=yλ and x=xλ. Then, the sequence <ynλ>α-converges to yλ and xλ∀λ∈∧. We assume that there is a μ∈∧ such that ynμ does not α-converges to yμ. Then, there is a σμ−α-open set Uμ containing yμ such that ynμ is not eventually in Uμ. We consider the set U=Πλ∈∧Yλ×Uα. Then, U is a α-open subset of Y by lemma in [7] and y∈U. Also, yn does not eventually converges to U, despite the fact that <yn>α-converges to y. As a result, we receive ynλ. For any λ∈∧, where ∧ is the index set, α-converges to yλ and xλ. Because each λ∈∧ is α-US, Yλ is α-US. As a result, x=y. As a result, Y is α-US.


## 6. Sequentially Sub-α-continuity and Main Results

This section introduces and investigates the ideas of sequentially subα-continuity, sequentially nearly α-continuity, and sequentially α-compact preserving functions, as well as their relationships and the property of α-US spaces.Definition 17.
The mappingξ:Y⟶X is named sequentially nearly α-continuous if for every point y∈Y, and every sequence yn in Yα-converges to y, and there is a subsequence ynk of yn such that <ξynk>⟶αξy.Definition 18.
The mappingξ:Y⟶X is named sequentially sub α-continuous if for every point y∈Y, and every sequence yn in Yα-converges to y, there is a subsequence ynk of yn, and a point x∈X such <ξynk>⟶αx.Definition 19.
If the imageξK of every sequentially α-compact set K of Y is sequentially α-compact in X, the mapping ξ:Y⟶X is said to be sequentially α-compact preserving.Lemma 8.
Every functionξ:Y⟶X is sequentially sub-α-continuous if X is sequentially α-compact.Proof.
We allowyn to be a sequence in which Yα-converges to y of Y. Then, ξyn is a sequence in X, and since X is sequentially α-compact, there exists a ξynk subsequence of ξynα− converging to a point x∈X. As a result, ξ:Y⟶X is sequentially sub- α− continuous.Theorem 14.
Each sequentiallyα-continuous is also sequentially continuous.Proof.
Letξ:Y⟶X be sequentially continuous and yn be a sequence in Y which α-converges to y∈Y. Then, yn converges to y since ξ is sequentially continuous. However, we are aware of this <yn>α-converging to y, so ξyn⟶αξy implies that ξ is sequentially α-continuous.Theorem 15.
Each function that is nearlyα-continuous function in the sequence is nearly α-compact in the sequence.Proof.
We assumeξ:Y⟶X is a sequentially nearly α-continuous function and K is any sequentially α-compact subset of Y. We will demonstrate that ξK is sequentially nearly α-compact set of X. Let xn represent any sequence in ξK. Then, for every positive integer n, there exists a point yn∈K with the formula ξyn=xn. Because yn is a sequence in the sequentially α-compact set K, there exists a sequence ynk of <yn>α-converging to y∈K. According to hypothesis, ξ is sequentially nearly α-continuous, and thus, there is a subsequence yj of ynk such that ξyj⟶αξy. As a result, there is a subsequence xj of <xn>α-converging to ξy∈ξK. This demonstrates that in X, ξK is sequentially α-compact set.Theorem 16.
Each sequentiallyα-compact preserving function is sequentially sub α-continuous.Proof.
Letξ:Y⟶X be a sequentially α-compact preserving function that is sequentially α-compact. Let y be any point in Y, and yn be any Yα-sequence converges to y. B and K=B∪y will be used to represent the set yn|n=1,2,3,…. Since yn⟶αy is sequentially α-compact, we get that K is sequentially α-compact. According to the hypothesis, ξ is a sequentially α-compact set of X, and ξK is a sequentially α-compact set of X. Because ξyn is a sequence in ξK, there is a ξynk subsequence of ξyn that α-converges to a point x∈ξK.
This signifies thatξ is sub α-continuous in a sequential order.Theorem 17.
If and only ifξ|K:K⟶ξyn is sequentially sub α-continuous for each sequentially α-compact subset K of Y, the function ξ:Y⟶X is sequentially α-compact preserving.Proof.
On the one hand, we assume thatξ:Y⟶X is a sequentially α-compact function. Then, ξK is α-compact sequentially set in X for each sequentially α-compact set K of Y. As a result, according to Lemma 8, ξ|K:K⟶ξK is sequentially a α-continuous function.
On the other hand, letK be any set of Y that is sequentially α-compact, and we will show that in X, ξK is a sequentially α-compact set. Let us say that xn be any sequence in ξK that can be used. After that, there is a point yn∈K where ξyn==xn for each positive integer n. A subsequence ynk of <yn>α-converging to a point y∈K exists because yn is a sequence in the sequentially α-compact set K. According to the hypothesis, ξ|K:K⟶ξK is sequentially sub α-continuous, so there is a subsequence xnk of <xn>α-converging to a point x∈ξK. This means that ξK is a α-compact set in X. As a result, ξ:Y⟶X is a sequentially α-compact preserving function.
A necessary requirement for a sequentially subα-continuous function to be sequentially α-compact preserving is the following corollary.Corollary 2.
If a functionξ:Y⟶X is sequentially sub α-continuous and ξK is a sequentially α-closed set in X for each sequentially α-compact set K of Y, then ξ is a sequentially α-compact preserving function.Proof.
Obvious


## 7. Conclusion and Future Work

In point-set-topology, very regular spaces will yield several novel topological features (such as separations axioms, compactness, connectedness, and continuity) which have been shown to be highly valuable in the study of specific objects of digital topology [15]. As a result, we may emphasize the relevance of almost α-completely regular spaces as a source of them, as well as their potential uses in computer graphics and quantum physics [6]. α-completely regular and almost α-completely regular spaces can also be studied in the bitopological space, and the characteristics of expanding this topological space can be studied in future work.



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