α-Completely Regular and Almost α-Completely Regular Spaces

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. If (Y, 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.


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 ζ z [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.
If (Y, 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.
for each open set V of X, the function ξ: Y ⟶ X is called α-continuous [4].

α-Zero Sets
In this section, we introduce α-zero sets and look into some of its properties.
Remark 3. In a space Y, if ξ: Y ⟶ R is an α-continuous function, then the set y ∈ Y: ξ(y) � 0 is an α-zero set.

Lemma 1.
If Y 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.
is a finite family of α-continuous functions, then the functions M, m: If a ∈ R and ξ: Y ⟶ R is an α-continuous function, then the sets B � y ∈ Y: ξ(y) ≥ a and D � y ∈ Y: ξ(y) ≤ a are α-zero sets in Y (vi) If a ∈ R and ξ: X ⟶ R is an α-continuous function, then the sets B � y ∈ Y: ξ(y) < a and D � y ∈ Y: ξ(y) > a are co-α-zero sets in Y ese results can be proved by using Lemma 4 and 5 (see [10], p.18). Theorem 1. If B 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.
e second part is proved similarly.

Characterizations of α-Completely Regular Spaces
We start three section with some characterizations of. Z If B � y for some y ∈ Y, then V in the Definition 6 is the neighbourhood of the point y.
e proof is obvious.

Characterizations of Almost α-Completely Regular Spaces
In this section, we are introducing to characterize the almost α-completely regular spaces using the α-zero sets and coα-zero sets in the following theorem. We recall that a point y ∈ Y is called a δ-accumulation point of a subset B of Y [11] at B⊓U ≠ ϕ for every open set U⊒y. e δ closure of B is indicated by C δ (B) and consists of all δ-accumulation points of B. e set B is named δ closed [11,12] e family of all open sets on Y is a topology, coarser than ζ denoted by ζ δ .
Complements of δ-closed sets are called δ open, and the family of all δ-open sets on Y is a topology.

α-US-Spaces and Results
Definition 8. A sequence 〈y n 〉 is called α-converges to a point y of Y, written as <y n > ⟶ α y if 〈y n 〉 appears in each α-open set containing y at some point. Any sequence <y n > α-converges to a y point; therefore, 〈y n 〉 converges to y is obvious.
e Y space is named α-US if each sequence 〈y n 〉 in Yis the value of α-converges to a single point.
We remember that space Y is known as US-space [13] if every convergent sequence converges to exactly one limit point.

Corollary 1. Every α-US-Space is US-Space and vice versa is not always true.
Example 1. If R 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. is demonstrates that not every US space is alpha-US.

Theorem 6. Every α-US-Space is α
Proof. We allow Y to be α-US Space. Let y and x be two separate points of Y. We consider the sequence 〈y n 〉, where y n � y for every n. Clearly, <y n > α-converges to y. Also, since y ≠ x and Y is α-US, 〈y n 〉 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 〈x n 〉 where x n � x for every n and proceed as previously. As a result, the Y space is α − T 1 . □ Theorem 7. Every α − T 2 space is α-US.
Proof. Let Y denote α − T 2 -space and 〈y n 〉 denote a Y sequence. We assume that <y n > α-converges to two separate points y and x if possible. at is, <y n > appears in every α-open set containing y, as well as every α-open set that contains x. Because Y is α − T 2 space, that is a contradiction. As a result Y is α-US. □ Definition 10. If each sequence in E α-converges to a point in E, the set E is sequentially α closed.
Proof. Let Y be α-US and 〈y n , y n 〉 be a Δ sequence. e sequence Y is then 〈y n 〉. Because Y is α-US, <y n > ⟶ α y for a unique y ∈ Y, implying that <y n > α-converges to y and x. As a result, x � y. As a result, Δ is an α-closed set.
We allow Δ to be α-closed consecutively and allow <y n > α-converging to y and x by allowing the sequence 〈y n 〉 to converges to y and x. As a result, the sequence <y n , y n > 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. e G 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 sequentially α-compact set in α-US-space is sequentially α-closed.
Proof. Let Y represent α-US-space and X represent a sequentially α-compact subset of Y. Let 〈y n 〉 be a X sequence. We assume that <y n > α-converges to a point in Y − X. Because X is sequentially α-compact, let 〈y np 〉 be a subsequence of 〈y n 〉 that α-converges to a point x ∈ X. In addition, we suppose a subsequence of 〈y np 〉 of <y n > α-converging to a point y in Y − X, x � y because 〈x np 〉 is a sequence in the α− US space Y. As a result, X is a sequentially α− closed set.
After that, we provide α-US space hereditary properties.

□ Theorem 10. An α-US-α subset spaces is α-US.
Proof. Let X⊑Y be a α-set and Y be α-US-space. Let 〈y n 〉 be a sequence in X. Let us pretend that <y n > α-converges to y, x in X. We will show that <y n > 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.
en, according to Lemma 2.16 [4], U⊓X and V⊓X are α-open sets in X. As a result, 〈y n 〉, as well as U and V, are finally in U⊓X and V⊓X. is means x � y because Y is α-US. As a result, the X subspace is α-US. Proof. Let Y be α − T 2 of a space. Hence, Y is α − R 1 by lemma in [7] and α-US by eorem 5.4. Conversely, let Y be both α − R 1 and α-US space. en, to prove that Y is α − T 2 , we have by eorem 5.3 that above every α-US space is α − T 1 , and Y is both α − T 1 and α − R 1 , and it follows from lemma in [7] that space Y is α − T 2 .  e α-point in the sequence 〈y n 〉 is the x point, if x is a <y n > α-cluster point. However, there is no 〈y n 〉 subsequence, the value of α converges to x. Definition 14. Each sequence that <y n > α-converges with a subsequence of <y n > α-side points and the Y topological space is said to be α − S 1 if it is α-US. Definition 15. e Y topological space is said to be α − S 2 if it is α-US, and every sequence 〈y n 〉 in Y α-converges which has no α-side points.
Obviously, by definition it follows that the results: In the subsequent, we have defined sequentially α-continuous functions based on the concept of sequentially continuous functions. Definition 16. If ξ(y n )α-converges to ξ(y) whenever 〈ξ n 〉, 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.
Proof. We assume X is α-US, and there is a sequence 〈y n 〉 in B that is α-converging to y ∈ Y. Because ξ and η are sequentially α-continuous functions in the same order, then ξ(y n ) ⟶ a ξ(y) and η(y n ) ⟶ α η(y). us, ξ(y) � η(y) and y ∈ B. As a result, B is sequentially α-closed. Now, we are going to show that α-US spaces have a product theorem. □ Theorem 13. α-US is the product of the arbitrary family of α-US spaces.
Let the sequence 〈y n 〉 in Y α-converges to y � (y λ ) and x � (x λ ). en, the sequence <y nλ > α-converges to y λ and x λ ∀ λ ∈∧. We assume that there is a μ ∈∧ such that 〈y nμ 〉 does not α-converges to y μ . en, there is a σ μ − α-open set U μ containing y μ such that 〈y nμ 〉 is not eventually in U μ . We consider the set U � Π λ∈∧ Y λ × U α . en, U is a α-open subset of Y by lemma in [7] and y ∈ U. Also, 〈y n 〉 does not eventually converges to U, despite the fact that <y n > α-converges to y. As a result, we receive 〈y nλ 〉. For any λ ∈∧, where ∧ is the index set, α-converges to y λ and x λ . Because each λ ∈∧ is α-US, Y λ is α-US. As a result, x � y. As a result, Y is α-US.

Sequentially Sub-α-continuity and Main Results
is section introduces and investigates the ideas of sequentially sub α-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.
e mapping ξ: Y ⟶ X is named sequentially nearly α-continuous if for every point y ∈ Y, and every sequence 〈y n 〉 in Y α-converges to y, and there is a subsequence 〈y nk 〉 of 〈y n 〉 such that <ξ(y nk ) > ⟶ α ξ(y). Definition 18. e mapping ξ: Y ⟶ X is named sequentially sub α-continuous if for every point y ∈ Y, and every sequence 〈y n 〉 in Y α-converges to y, there is a subsequence 〈y nk 〉 of 〈y n 〉, and a point x ∈ X such <ξ(y nk ) > ⟶ α x.
Definition 19. If the image ξ(K) 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.
Proof. We allow 〈y n 〉 to be a sequence in which Y α-converges to y of Y. en, ξ(y n ) is a sequence in X, and since X is sequentially α-compact, there exists a ξ(y nk ) subsequence of ξ(y n ) α− converging to a point x ∈ X. As a result, ξ: Y ⟶ X is sequentially sub-α− continuous. □ Theorem 14. Each sequentially α-continuous is also sequentially continuous.
Proof. Let ξ: Y ⟶ X be sequentially continuous and 〈y n 〉 be a sequence in Y which α-converges to y ∈ Y. en, 〈y n 〉 converges to y since ξ is sequentially continuous. However, we are aware of this <y n > α-converging to y, so ξ(y n ) ⟶ α ξ(y) implies that ξ is sequentially α-continuous. □ Theorem 15. Each function that is nearly α-continuous function in the sequence is nearly α-compact in the sequence.
Proof. We assume ξ: Y ⟶ X is a sequentially nearly α-continuous function and K is any sequentially α-compact subset of Y. We will demonstrate that ξ(K) is sequentially nearly α-compact set of X. Let 〈x n 〉 represent any sequence in ξ(K). en, for every positive integer n, there exists a point y n ∈ K with the formula ξ(y n ) � x n . Because 〈y n 〉 is a sequence in the sequentially α-compact set K, there exists a sequence 〈y nk 〉 of <y n > α-converging to y ∈ K. According to hypothesis, ξ is sequentially nearly α-continuous, and thus, there is a subsequence 〈y j 〉 of 〈y nk 〉 such that ξ(y j ) ⟶ α ξ(y). As a result, there is a subsequence 〈x j 〉 of <x n > α-converging to ξ(y) ∈ ξ(K). is demonstrates that in X, ξ(K) is sequentially α-compact set. □ Theorem 16. Each sequentially α-compact preserving function is sequentially sub α-continuous.
Proof. Let ξ: Y ⟶ X be a sequentially α-compact preserving function that is sequentially α-compact. Let y be any point in Y, and 〈y n 〉 be any Y α-sequence converges to y. B and K � B ∪ y will be used to represent the set y n |n � 1, 2, 3, . . . . Since y n ⟶ α y 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 ξ(K) is a sequentially α-compact set of X. Because ξ(y n ) is a sequence in ξ(K), there is a ξ(y nk ) subsequence of ξ(y n ) that α-converges to a point x ∈ ξ(K).
is signifies that ξ is sub α-continuous in a sequential order. □ Theorem 17. If and only if ξ|K: K ⟶ ξ(y n ) 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 that ξ: Y ⟶ X is a sequentially α-compact function. en, ξ(K) 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 ⟶ ξ(K) is sequentially a α-continuous function.
On the other hand, let K be any set of Y that is sequentially α-compact, and we will show that in X, ξ(K) is a sequentially α-compact set. Let us say that 〈x n 〉 be any sequence in ξ(K) that can be used. After that, there is a point y n ∈ K where ξ(y n ) �� x n for each positive integer n. A subsequence 〈y nk 〉 of <y n > α-converging to a point y ∈ K exists because 〈y n 〉 is a sequence in the sequentially α-compact set K. According to the hypothesis, ξ|K: K ⟶ ξ(K) is sequentially sub α-continuous, so there is a subsequence 〈x nk 〉 of <x n > α-converging to a point x ∈ ξ(K). is means that ξ(K) 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 sub α-continuous function to be sequentially α-compact preserving is the following corollary. □ Corollary 2. If a function ξ: Y ⟶ X is sequentially sub α-continuous and ξ(K) 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.

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.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.