# Functionally Separation Axioms on General Topology

Authors: Abdelwaheb Mhemdi; Tareq M. Al-shami
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we define a new family of separation axioms in the classical topology called functionallyTi spaces for i=0,1,2. With the assistant of illustrative examples, we reveal the relationships between them as well as their relationship with Ti spaces for i=0,1,2. We demonstrate that functionally Ti 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 Sierpinski spaces.


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## Body

## 1. Introduction and Preliminaries

Topology is a branch of mathematics that researches properties of spaces that are invariant under any continuous deformation. That 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 [1–3].Separation axioms are one among the most common, important, and interesting ideas in topology. They 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, [4–6]) 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 [8–11]). 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]).This work is organized as follows: after this introduction, we recall some definitions and results which are required to make this work self-contained. In Section2, we introduce the concepts of functionally Ti 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.The following definitions are mentioned in [15].Definition 1.
A subcollectionτ of 2X is called a topology on X provided that it is closed under arbitrary union and finite intersection. A pair X,τ is called a topological space. Every subset of X belong to τ is called an open set, and its complement is called a closed set.Definition 2.
A bijective mapf:X,τ⟶Y,θ is said to be a homeomorphism if it is continuous (i.e., the inverse image of every open (resp., closed) set is open (resp., closed)) and open (i.e., the image of every open set is open).
A property which is preserved by every homeomorphism map is called a topological property.Definition 3.
LetA be a set in a topological space X,τ. The family τA=A∩G:G∈τ is called a relative topology on A. A pair A,τA is called a subspace of X,τ.
A property which is preserved by every subspace is called a hereditary property.Definition 4.
X,τ is said to be as follows:(1)
T0 provided that, for every a≠b∈X, there exists an open set containing only one of them(2)
T1 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)
T2 (or Hausdorff) provided that there exist two disjoint open sets U and V for every a≠b∈X such that a∈U and b∈VProposition 1.
For any two setsAandB, we haveG×H∩U×V=G∩H×U∩V.Definition 5.
LetXi,τi:i=1,2,…,n be the collection of topological spaces. Then, ℬ=∏i=1nτi=∏i=1nGi:Gi∈τi defines a base for a topology τ on X=∏i=1nXi. We call X,τ a finite product space.Definition 6.
A binary relationρ on X (i.e., ρ⊆X×X) is said to be transitive if x,z∈ρ whenever x,y∈ρ and y,z∈ρ.


## 2. Relative Separation Axioms

In this portion, we formulate the concepts of functionallyTi 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.
LetX,τ,Y,θ be two topological spaces. X,τ is said to be as follows:(1)
A functionallyT0 space relatively to Y,θ or, for short, X is FT0,Y if for any distinct points x,y∈X, there exist a continuous map f from X to Y and U∈θ such that fx,fy∩U∈fx,fy(2)
A functionallyT1 space relatively to Y,θ or, for short, X is FT1,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 fx∈U/V and fy∈V/U(3)
A functionallyT2 space relatively to Y,θ or, for short, X is FT2,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 U∩V=∅, fx∈U and fy∈VRemark 1.
(1)
IfX is a singleton, then for any topological space Y,θ, X is FTi,Y for each i=0,1,2(2)
IfY,θ is the indiscrete topology and X≥2, then X,τ is not FT0,Y for each topology τ on XProposition 2.
X,τis aTi-space if and only if it isFTi,Xfor eachi=0,1,2.Proof.
It is sufficient to take the identity mapid 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.Proposition 3.
EveryFTi,Yspace isFTi−1,Yfor eachi=1,2.We give the following example to clarify the fail of the converse of Proposition3.Example 1.
Letτ1=∅,X,a,a,b be a topology on X=a,b,c and τ2 be the cofinite topology on the set of natural numbers ℕ. It is clear that X,τ1 and ℕ,τ2 are, respectively, T0 and T1 spaces. On the other hand, X,τ1 is not a T1 space and ℕ,τ2 is not a T2 space. Without loss of generality, consider X=Y. Then, it follows from Proposition 3 that X,τ1 is FT0,Y but not FT1,Y. Also, consider Y=ℕ. Then, ℕ,τ2 is FT1,Y but not FT2,Y.Proposition 4.
For alli∈0,1,2, ifX,τisFTi,Y, thenX,τis aTi-space.Proof.
Wheni=0. Suppose that x≠y in an FT0,Y-space. Then, there exist a continuous map f from X to Y and U∈θ such that fx,fy∩U∈fx,fy. Now, f−1U is an open subset of X containing only one of the elements x and y but not both. Hence, X is a T0-space.
By analogous it is easy to prove the proposition fori∈1,2.Corollary 1.
IfX,τis a finiteFT1,Yspace, thenτis the discrete topology.The converse of the proposition above fails as shown in the following example.Example 2.
Letτ be the discrete topology on X=a,b,c and θ=∅,Y,1 be a topology on Y=1,2,3,4. It is clear that X,τ is a T2 space. On the other hand, let f:X,τ⟶Y,θ be any continuous map. While fa, fb and fc are three distinct points in Y,θ, a topology θ contains only one proper open subset of Y; hence, X,τ is not FT2,Y.
We summarize the previous relationships in Figure1.Figure 1
The relationships between functionally separation axioms and classical separation axioms.Theorem 1.
IfX,τandY,θareTispace such thatXis finite, thenX,τis anFTi,Y-space, wherei=1,2.Proof.
We only prove the theorem in case ofi=1.
SinceX,τ a finite T1 space, then every map from X to Y is continuous. Therefore, for every x≠y∈X, we define a continuous map f such that fx≠fy∈Y. Since Y,θ is a T1 space, there exist open subsets U,V of Y such that fx∈U/V and fy∈V/U. Hence, the proof is complete.
In the following result, we investigate under which conditions the converse of Proposition4 is true.Proposition 5.
Letg:X,τ⟶Y,θbe an injective bicontinuous map. Then,X,τis aFTi,Yspace if and only ifX,τis aTispace for eachi=0,1,2.Proof.
We only prove the proposition incase ofi=2. The other cases are proved similarly.
The necessary part follows from Proposition4. 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 gx≠gy∈Y, and since g is open, then gU and gV are disjoint open sets in θ containing gx and gy, respectively. It comes from the continuity of g that X,τ is a FT2,Y. Hence, we obtain the desired result.Theorem 2.
The finite product ofFTi,Yspaces isFTi,Yfor eachi=0,1,2.Proof.
We only prove the proposition incase ofi=2.
LetX1,τ1 and X2,τ2 be two FT2,Y spaces. Then, for all a1≠a2∈X1, there exists a continuous map f from X1 to Y and F,G∈τ1 such that F∩G=∅,fa1∈F and fa2∈G, and for all b1≠b2∈X2, there exists a continuous map g from X2 to Y and U,V∈τ2 such that U∩V=∅,fb1∈U and fb2∈V. Suppose that a1,b1≠a1,b1∈X1×X2. Then, a1≠a2 or b1≠b2. Without loss of generality, consider a1≠a2. Taking a map h:X1×X2⟶Y1×Y2 defined by ha,b=fa,gb. It can be checked that h is a continuous map. Now, we have F×Y2 is an open subset of Y1×Y2 containing ha1,b1 and G×Y2 is an open subset of Y1×Y2 containing ha2,b2 such that F×Y2∩G×Y2=∅. Hence, X1×X2 is a FT2,Y1×Y2.
By analogous one can prove the other two cases.Theorem 3.
The property of being anFTi,Aspace is a topological property for eachi=0,1,2.Proof.
We only prove the proposition incase ofi=2.
Suppose thatf:X,τ⟶Y,θ is a homomorphism map. To prove that Y,θ is a FT2,A space, let x≠y∈Y. Then, f−1x≠f−1y∈X. Since X,τ is a FT2,A space, then there exists a continuous map g from X to A and F,G∈τ such that F∩G=∅, gf−1x∈F and gf−1y∈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 FT2,A space. Hence, the proof is complete.
By analogous one can prove the other two cases.Theorem 4.
The property of being anFTi,Aspace is a hereditary property for eachi=0,1,2.Proof.
The proof follows from the fact a continuous mapf:X,τ⟶Y,θ implies that the restriction map f:A,τA⟶Y,θ is also continuous for every nonempty subset A of X.Theorem 5.
The binary relation defined on the set of topological spaces byX,τ∼Y,θif and only ifXisFTi,Yis transitive.Proof.
LetX,τ,Y,θ, and Z,τZ be three topological spaces such that X,τ∼Y,θ and Y,θ∼Z,τZ. 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 fx≠fy, 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∘fx and g∘fy. Thus, X,τ∼Z,τZ; hence, ∼ is transitive.


## 3. Properties and Particular Cases

In this section, we present some properties ofFTi,Y spaces under some particular of topological spaces such as discrete and Sierpinski spaces.Proposition 6.
LetX,τandY,θbe two topological spaces such thatX>1andY=2. IfX,τisFT1,Y, thenτandθare discrete topologies.Proof.
Letx≠y∈X. Then, there exists a continuous map from X to Y and U∈θ such that fx∈U and fy∉U, then U=fx∈θ. Similarly, we prove that fy∈θ. Thus, θ=τd. Also, it follows from Corollary 1 that τ is the discrete topology on X.
In the rest of this paper we will denote byD the topological space 0,1 with the discrete topology.Theorem 6.
The next statements are identical:(1)
X,τ is FT0,D(2)
For allx≠y, there exists a clopen (open and closed) containing x and not containing yProof.
1⇒2: suppose X be FT0,D. Let x≠y∈X, then there exists f∈CX,Y such that fx=0 and fy=1. So that f−1fx is a clopen containing x but not containing y.2⇒1: let x≠y∈X and U a open set containing x and not containing y. Now, the map f from X to D defined by fU=0 and fUc=1 is a continuous map. Now, we have fU is an open neighborhood of fx such that fy∉fU, as required.
Using Theorem6, we can deduce the next corollary.Corollary 2.
We have the following equivalences:(1)FT0,D⇔FT1,D⇔FT2,D.
Now, we denote byS the Sierpinski space S=0,1 with the topology τS=∅,S,1.Theorem 7.
LetX,τbe a topological space such thatXis not a singleton. Then, the next statements are identical:(1)
X is an FT0,S-space(2)
X is a T0 spaceProof.
1⇒2: by Proposition 4.2⇒1: suppose X is a T0-space. Let x≠y∈X. Then, there is open set U containing x or y but not both. Say, x∈U. Now, the map f from X to S defined by fU=1 and fUc=0 is continuous and 1 is an open subset containing fx and not containing fy. This fact completes the proof.Remark 2.
There is no topological space which isFT1,S, so that there is no one which is FT2,S.
We denote byOa the set of all open neighborhoods of a in a topological space X,τ.Theorem 8.
The next statements are identical:(1)
S is FT0,Y(2)
∃a,b∈Y,θ such that Ob⊂Oa strictlyProof.
1⇒2: let f be a continuous map from S to Y and U∈θ such that f1∈U and f0∉U. Then, Of0⊂Of1 strictly.2⇒1: let a,b∈Y such that Ob⊂Oa strictly. We define the map f from S to Y by f1=a and f0=b. On the one hand, f is continuous because every open set containing b contain also a. On the other hand, since the inclusion Ob⊂Oa is strict, there exists an open neighborhood of f1 which does not contain f0; hence, we deduce that S is FT0,Y.Corollary 3.
IfSisFT0,Y, thenYis not aT1-space.
LetX,τ be a Functionally Hausdorff space. Then, X,τ can be seen as FT2,ℝ. What about a topological space Y,θ such that ℝ is FT2,Y. The following proposition gives a sufficient condition for such topological space.Proposition 7.
LetY,θbe a topological space. IfY,θsatisfies the following conditions:
∃a,b,c∈Y,U∈OaandV∈Obsuch thatU∩V=∅andOc⊆Oa∩Ob.
Then,ℝ is FT2,Y.Proof.
Letx<y∈ℝ and f be the map from ℝ to Y defined by f−∞,x+y/2=a, fx+y/2,−∞=b and fx+y/2=c. Then, f satisfies the third separation axioms in Definition 7.
The following example shows that the condition in the previous proposition is sufficient but not necessary for a topological spaceY,θ such that ℝ is FT2,Y.Example 3.
LetY=1,2,3 with the topology θ=∅,1,2,1,2,Y. Then, ℝ is FT2,Y. Note that ℝ does not satisfy the condition given in the above proposition because for any three elements a=1,b=2,c=3ofℝ, we have O1⊆O2∩O3, O2⊆O1∩O3, and O3⊆O1∩O2. On the other hand, ℝ is FT2,ℝ.
It is natural now to pose the question about the necessary and sufficient condition for a topological spaceY,θ such that ℝ is FT2,Y. The following theorem answers this question.Theorem 9.
LetY,θbe a topological space. Then, the next properties are identical:(1)
ℝ is FT2,Y(2)
∃U,V∈θ and a continuous map g from ℝ to Y such that U∩V=∅, g0∈U and g1∈VProof.
The first implication is trivial by the definition ofFT2,Y.
Conversely, ifx and y are two distinct real numbers, then there exists an isomorphism h from ℝ to itself satisfying hx=0 and hy=1. So that the map f=g∘h is continuous such that fx∈U and fy∈V. This completes the proof.
It is well known that any map from a discrete topological spaceX,τd to any topological space Y,θ is continuous. We benefit from this fact to establish the following results.Proposition 8.
LetY,θ be a topological space and X≥2. Then, the following properties are equivalent:(1)
X,τd is FT0,Y(2)
θ≥3Proof.
1⇒2: this direction is obvious.2⇒1: let a≠b∈X and O∈θ/Y,∅. We define a map (continuous) f from X to Y by fa∈O and fx∉O for all x≠a. Hence, X,τ is FT0,Y, as required.Proposition 9.
LetY,θbe a topological space andX≥2. Then, the following properties are equivalent:(1)
X,τd is FT1,Y(2)
There exist two distinct pointsy,z∈Y such that the topology of the subspace A=y,z is the discrete topologyProof.
Suppose thatX is FT1,Y and let a≠b∈X. Then, there exists U,V∈θ such that fx∈U/V and fy∈V/U. It is clear that the subspace A=fa,fb is equipped by the discrete topology.
Conversely, it is sufficient, for every two distinct pointsa,b in X, a map f such that fa=y and fb=z.Theorem 10.
LetY,θbe a topological space andX≥2. Then, the following properties are equivalent:(1)
X,τd is FT2,Y(2)
There existO1,O2∈θ such that O1∩O2=∅Proof.
1⇒2: this direction follows from 3 of Definition 7.2⇒1: let a≠b∈X. We define a (continuous) map f from X to Y such that fa∈O1 and fx∉O2 for all x≠a. Hence, we obtain that X,τd is FT2,Y, as required.Corollary 4.
IfY,θ is disconnected topological space, then X,τd is FTi,Y for each i∈0,1,2.


## 4. Conclusion

This manuscript contributes to the area of separation axioms. We have applied continuous maps to introduce the concepts of functionallyTi spaces for i=0,1,2. We have elucidated the relationships between them and clarified that they are weaker than Ti 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].



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