Functionally Separation Axioms on General Topology

In this paper, we deﬁne a new family of separation axioms in the classical topology called functionally T i spaces for i � 0 , 1 , 2. With the assistant of illustrative examples, we reveal the relationships between them as well as their relationship with T i spaces for i � 0 , 1 , 2. We demonstrate that functionally T i spaces are preserved under product spaces, and they are topological and hereditary properties. Moreover, we show that the class of each one of them represents a transitive relation and obtain some interesting results under some conditions such as discrete and


Introduction and Preliminaries
Topology is a branch of mathematics that researches properties of spaces that are invariant under any continuous deformation.
at is, it forms a novel kind of geometry that relies on nearness or neighborhood of elements instead of measuring distance between them. Recently, topological spaces have been applied to model practical problems, especially those related to information system; see, for example [1][2][3].
Separation axioms are one among the most common, important, and interesting ideas in topology. ey can be used to initiate different classes of topological spaces and determine the type of some subsets. Some separation axioms were defined using continuous maps (see, [4][5][6]) on one hand. On the other hand, some separation axioms were introduced using generalized open sets (see [7] and the references mentioned therein). Also, separation axioms have been generalized to other spaces in general topology like pretopological spaces, ordered topological spaces, and generalized topological spaces (see [8][9][10][11]). Another generalization of separation axioms was given using functions in the category of topological spaces (see [12]). We draw attention to some separation axioms have been very useful in the study of certain objects in digital topology (see [13,14]). is work is organized as follows: after this introduction, we recall some definitions and results which are required to make this work self-contained. In Section 2, we introduce the concepts of functionally T i space for i � 0, 1, 2. We study their main properties, especially those are related to product spaces and topological and hereditary properties. In Section 3, we explore some findings that associated these concepts with the discrete and Sierpinski spaces. We write some conclusions and propose some future works in Section 4. e following definitions are mentioned in [15].
A property which is preserved by every subspace is called a hereditary property.
Definition 4. (X, τ) is said to be as follows: (1) T 0 provided that, for every a ≠ b ∈ X, there exists an open set containing only one of them (2) T 1 provided that, for every a ≠ b ∈ X, there exist two open sets such that one containing a but not b and the other containing b but not a (3) T 2 (or Hausdorff) provided that there exist two disjoint open sets U and V for every a ≠ b ∈ X such that a ∈ U and b ∈ V Proposition 1. For any two sets A and B, we have . . , n be the collection of topological spaces. en, B � n i�1 τ i � n i�1 G i : G i ∈ τ i defines a base for a topology τ on X � n i�1 X i . We call (X, τ) a finite product space.

Relative Separation Axioms
In this portion, we formulate the concepts of functionally T i space (i � 0, 1, 2) using continuous maps. We explore their basic properties and provide some interesting examples to clarify the presented results. Definition 7. Let (X, τ), (Y, θ) be two topological spaces. (X, τ) is said to be as follows: (1) A functionally T 0 space relatively to (Y, θ) or, for short, X is FT (0,Y) if for any distinct points x, y ∈ X, there exist a continuous map f from X to Y and short, X is FT (1,Y) if for any distinct points x, y ∈ X, there exist a continuous map f from X to Y and U, V ∈ θ such that f(x) ∈ (U/V) and f(y) ∈ (V/U) (3) A functionally T 2 space relatively to (Y, θ) or, for short, X is FT (2,Y) if for any distinct points x, y ∈ X, there exist a continuous map f from X to Y and (1) If X is a singleton, then for any topological space Proof. It is sufficient to take the identity map id from (X, τ) to itself. It is clear that id is a continuous map satisfying the desired result. It is easy to prove the following implications.
We give the following example to clarify the fail of the converse of Proposition 3.
} and τ 2 be the cofinite topology on the set of natural numbers N. It is clear that (X, τ 1 ) and (N, τ 2 ) are, respectively, T 0 and T 1 spaces. On the other hand,

Proposition 4. For all
is an open subset of X containing only one of the elements x and y but not both. Hence, X is a T 0 -space.
By analogous it is easy to prove the proposition for e converse of the proposition above fails as shown in the following example.
We summarize the previous relationships in Figure 1.
Proof. We only prove the theorem in case of i � 1.
Since (X, τ) a finite T 1 space, then every map from X to Y is continuous. erefore, for every x ≠ y ∈ X, we define a In the following result, we investigate under which conditions the converse of Proposition 4 is true.
Proof. We only prove the proposition incase of i � 2. e other cases are proved similarly. e necessary part follows from Proposition 4. To prove the sufficient part, let x ≠ y ∈ X. By hypothesis, there exist two disjoint open sets U, V ∈ τ such that x ∈ U and y ∈ V. Since g is injective, we obtain g(x) ≠ g(y) ∈ Y, and since g is open, then g(U) and g(V) are disjoint open sets in θ containing g(x) and g(y), respectively. It comes from the continuity of g that (X, τ) is a FT (2,Y) . Hence, we obtain the desired result. Proof. We only prove the proposition incase of i � 2.
Let (X 1 , τ 1 ) and (X 2 , τ 2 ) be two FT (2,Y) spaces. en, for all a 1 ≠ a 2 ∈ X 1 , there exists a continuous map f from X 1 to Y and F, G ∈ τ 1 such that F ∩ G � ∅, f(a 1 ) ∈ F and f(a 2 ) ∈ G, and for all b 1 ≠ b 2 ∈ X 2 , there exists a continuous map g from X 2 to Y and U, V ∈ τ 2 such that en, a 1 ≠ a 2 or b 1 ≠ b 2 . Without loss of generality, consider a 1 ≠ a 2 . Taking a map h: g(b)). It can be checked that h is a continuous map. Now, we have By analogous one can prove the other two cases. Proof. We only prove the proposition incase of i � 2.
Suppose that f: then there exists a continuous map g from X to A and F, G ∈ τ such that F ∩ G � ∅, g(f −1 (x)) ∈ F and g(f −1 (y)) ∈ G. Since f is a homomorphism and g is continuous, then g ∘ f −1 : Y ⟶ A is a continuous map satisfying that (Y, θ) is a FT (2,A) space. Hence, the proof is complete. By analogous one can prove the other two cases.  Proof. Let (X, τ), (Y, θ), and (Z, τ Z ) be three topological spaces such that (X, τ) ∼ (Y, θ) and (Y, θ) ∼ (Z, τ Z ). If x ≠ y ∈ X, then there exists a continuous map f from X to Y satisfying the conditions given in Definition 2.1 for each i. Now, since f(x) ≠ f(y), then there exists also a continuous map g from Y to Z satisfying the same conditions, so that the composition g ∘ f is a continuous map from X to Z verifying the conditions in the separation i for the points (g ∘ f)(x) and (g ∘ f)(y).

Properties and Particular Cases
In this section, we present some properties of FT (i,Y) spaces under some particular of topological spaces such as discrete and Sierpinski spaces. Proposition 6. Let (X, τ) and (Y, θ) be two topological spaces such that |X| > 1 and |Y| � 2. If (X, τ) is FT (1,Y) , then τ and θ are discrete topologies.
Proof. Let x ≠ y ∈ X. en, there exists a continuous map from X to Y and U ∈ θ such that f(x) ∈ U and f(y) ∉ U, us, θ � τ d . Also, it follows from Corollary 1 that τ is the discrete topology on X.
In the rest of this paper we will denote by D the topological space 0, 1 { } with the discrete topology. □ Theorem 6. e next statements are identical: Using eorem 6, we can deduce the next corollary. □ Corollary 2. We have the following equivalences: (2,D) . (1) Theorem 7. Let (X, τ) be a topological space such that X is not a singleton. en, the next statements are identical:  ere is no topological space which is FT (1,S) , so that there is no one which is FT (2,S) .
We denote by O(a) the set of all open neighborhoods of a in a topological space (X, τ).

Theorem 8.
e next statements are identical: Let (X, τ) be a Functionally Hausdorff space. en, (X, τ) can be seen as FT (2,R) . What about a topological space (Y, θ) such that R is FT (2,Y) . e following proposition gives a sufficient condition for such topological space. Proposition 7. Let (Y, θ) be a topological space. If (Y, θ) satisfies the following conditions: en, R is FT (2,Y) .
e following example shows that the condition in the previous proposition is sufficient but not necessary for a topological space (Y, θ) such that R is FT (2,Y) .
en, R is FT (2,Y) . Note that R does not satisfy the condition given in the above proposition because for any three elements . On the other hand, R is FT (2,R) .
It is natural now to pose the question about the necessary and sufficient condition for a topological space (Y, θ) such that R is FT (2,Y) . e following theorem answers this question.
Theorem 9. Let (Y, θ) be a topological space. en, the next properties are identical: (1) R is FT (2,Y) (2) ∃U, V ∈ θ and a continuous map g from R to Y such that U ∩ V � ∅, g(0) ∈ U and g(1) ∈ V Proof. e first implication is trivial by the definition of FT (2,Y) .
Conversely, if x and y are two distinct real numbers, then there exists an isomorphism h from R to itself satisfying h(x) � 0 and h(y) � 1. So that the map f � g ∘ h is continuous such that f(x) ∈ U and f(y) ∈ V. is completes the proof.
It is well known that any map from a discrete topological space (X, τ d ) to any topological space (Y, θ) is continuous. We benefit from this fact to establish the following results. en, the following properties are equivalent:
en, the following properties are equivalent:

Conclusion
is manuscript contributes to the area of separation axioms. We have applied continuous maps to introduce the concepts of functionally T i spaces for i � 0, 1, 2. We have elucidated the relationships between them and clarified that they are weaker than T i spaces, see Proposition 3 and Proposition 4. Also, we have investigated some results associated them with product spaces and categories. Moreover, their behaviours with some special spaces such as discrete and Sierpinski spaces have been studied.
In the upcoming work, we plan to explore functionally regular and functionally normal spaces. Also, we generalize the presented concepts presented in this manuscript using somewhere dense sets and SD-continuous maps given in [7,16,17].

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare no conflicts of interest.