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Osvaldo Mendez, Liviu Horia Popescu and Emil Daniel Schwab

Abstract.In this paper we present new forms of the classical separation axioms on topological spaces. Our constructions generate a method to refine separation properties when passing to the quotient space and our results may be useful in the study of algebraic topological structures, such as topological groups and topological vector spaces.

M.S.C. 2000: 54D10, 54D15, 54H11, 54H13.

Key words: separation axioms, binary relations, topological spaces.

1 Introduction

The classical separation axioms have been generalized in several directions. In this work, we will investigate theR-separated spaces, introduced in [3]. There, the author introduced the so calledR-separation properties on a topological spaceX on which a binary relationRis defined. The main idea is to replace the identity relation in the classical separation properties with the relationR. So, for instance, X is said to be R-Hausdorff iff given two R-unrelated elements, say a and b, there exist R-disjoint neighborhoods ofaandb respectively (see [3]).

In Section 2, we briefly describe the main constructions and results in [3].

In Section 3 we present Kolmogorov’s first relation on a topological spaceX, which is an order relation on T0 (Kolmogorov) spaces. The main result in this section is

Theorem 3.3, which shows that classical separability can be expressed in terms of separability with respect to Kolmogorov’s relation.

In Section 4 we study separability with respect to Kolmogorov’s second relation on a topological space, which as it turns out, is an equivalence relation. The central result in this section is Theorem 4.6. This theorem is actually the main result of the paper. It refines the separation properties of certain classes of topological spaces.We also show that the quotient space of Kolmogorov’s second relation satisfies supplementary separation properties. We present concrete applications to topological groups and topological vector spaces.

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Balkan Journal of Geometry and Its Applications, Vol.13, No.2, 2008, pp. 59-65. c

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2 Notation and terminology

In consistency with the notation in [3], given a topological space (X, τ) together with a binary relation R on X we will denote by Vx the filter of neighborhoods of the

elementx∈X. For a set A⊆X, VA will stand for an open subset ofX containing A. As usual, the inverse ofRwill be written as R−1_{, whereas for} _x_∈_X _and _y_∈_X

the notationxRy will indicate that the pair (x, y) does not belong to the graph ofR. Analogously, for A⊆X andB ⊆X, we will write ARB to indicate that the cross productA×Bis contained in the complement of the graph ofR. We will abbreviate {x}RAasxRA. Forx∈X, the set{y:xRy}will be written asR(x). More generally, ifY ⊆X,R(Y) =S

x∈Y R(x).

The spaceX will be said to be:

• TR

0 iff wheneverx1∈X,x2∈X andx1Rx2, there existsV ∈ Vxj forj= 1 or

j= 2 such thatxiRV fori6=j.

• TR

1 iff wheneverx1∈X,x2∈X andx1Rx2, there existVi∈ Vxi,i= 1,2 such thatxiRVj fori6=j.

• TR

2 iff for x1 ∈X,x2∈X andx1Rx2, there existVi ∈ Vxi, i= 1,2 such that

V1RV2.

• TR

3 or R-regular iff for any closed subset F ⊂ X and x ∈ X, xRF implies

the existence of neighborhoodsVxandVF such thatVxRVF, andFRximplies VFRVxfor some neighborhoods Vx andVF.

• TR

4 orR-normal iff for each A ⊂ X and B ⊂ X such that ARB, there are

neighborhoodsVA andVB such thatVARVB.

In [3], characterization theorems for the above separation properties were shown, that reduce to classical equivalent properties of separation when the relation under consideration is the identity onX. For example,X isTR

1 if and only if the setsR(x)

andR−1₍_x_{) are closed.}

3 Kolmogorov’s first relation

In this section we investigate the previous ideas in the particular case of the relation

(3.1) xRy iff y∈ {x},

called Kolmogorov’s (first) relation (hereM denotes the closure of the setM). Notice that the above condition is equivalent to

(3.2) {y} ⊆ {x}.

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X).

We start by pointing out the trivial facts thatRis reflexive and transitive and that forx∈X, one has

R(x) ={x} and R−1₍_x_{) =} \

A∈V(x) A

whereV(x) denotes the filter of neighborhoods ofx. Moreover, it is not hard to verify that for any subsetAofX and any open subset Ω, it holds thatR(A) =R(A) =A

andR−1_{(Ω) = Ω. Notice also that any two closed (respectively, open) sets are disjoint}

if and only if they areR-disjoint.

Lemma 3.1. The separation axiomTR

0 holds for the Kolmogorov relationR onX.

Moreover,X isT0 if and only ifRis an order relation.

Proof. The first part is an immediate consequence of the definition of R. Next, observe thatRis an order relation if and only if it is antisymmetric. In that case, ifa

andbare two different elements ofX, then the fact that at least one of the statements

aRb or bRa is false, yields the validity of the T0 axiom immediately. The converse

follows easily. ✷

Lemma 3.2. If the spaceX isTR

1 , thenRis symmetric.

Proof. If X isTR

1 anda∈X,b ∈X with aRb, then the assumptionbRayields

the existence of a neighborhoodVb ofb withVbRa. This is impossible, sincea∈Vb.

✷

Theorem 3.3. Let X be a topological space and R stand for Kolmogorov’s first relation. Then the following statements are equivalent:

(i) X isTR

1 ;

(ii) Ris the identity relation on X;

(iii) X isT1.

Proof. The implication (i) ⇒ (ii) follows directly from Lemma 3.2 and the fact that, beingTR

1 ,X is alsoT

R

0 , and hence antisymmetric.

The statement (ii)⇒(iii) is an immediate consequence of the definition ofR. Finally, ifX isT1andx1Rx2, then by definition ofR, there is an open neighborhood V2 of x2 such that x1 ∈/ V2. It is clear that no element of V2 is in the closure of x1, which yields x1RV2. On the other hand, since x1 6= x2, we infer from (iii) the

existence of a neighborhoodV1 ofx1not containingx2, which immediately yields the

statementx2RV1. By definition,X isT1R. ✷

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Proof.IfX is regular,x∈X andF ⊂X is closed withxRF, it follows thatx /∈F. Accordingly, there are disjoint open setsVxandVF containingxandF respectively.

It follows now by definition ofRthatVxRVF. A similar reasoning shows that ifFRx,

one can findR-disjoint open neighborhoods ofF andx.

Conversely, assuming thatX isR-regular, given a closed subsetF ⊆X andx /∈F, we consider an open neighborhood ofx,Vx, disjoint withF. Clearly, FRx. LetWx

andWF be open neighborhoods ofxandF respectively such thatWFRWx. Then it

is clear thatWx∩WF =¡❢. ✷

Lemma 3.5. A topological spaceX is normal if and only if it isR-normal.

Proof. If X is normal and F and G are R-disjoint closed subsets of X, then

F∩G=¡❢_{. Let}_V_F _and_V_G_{be disjoint open neighborhoods of}_F _and_G_{respectively.}

It is clear thatVF andVG areR-disjoint.

Conversely, assume that X is R-normal and consider disjoint closed subsets F and

G. It is clear thatFRG. There are thenR-disjoint (hence , also disjoint) open neigh-borhoodsVF and VG ofF andGrespectively.

✷

Theorem 3.6. For eachi,1≤i≤4, the separation axiomsTiandTiRare equivalent.

Proof. The proof follows immediately from the above Lemmas. ✷

4 Kolmogorov’s second relation

LetX be a topological space. We define the relationρonX as follows:

(4.1) xρyif and only if{x}={y}

Notice thatρis an equivalence relation on X and thatxρy⇔xRy andyRx, where Rstands for Kolmogorov’s First Relation defined in the previous section.

We also point out the fact thatX is T0if and only ifρis the identity onX. Therefore,

setting ˆxto be the equivalence class ofx∈X, we have

(4.2) ˆx=R(x)∩ R−1₍_x₎

for eachx∈X. It is also clear that any topological space isT₀ρ and thatρ(Ω) = Ω for any open set Ω.

Therefore (see [3]) the following statement holds:

Theorem 4.1. IfX isT_iρ,1≤i≤4, then the quotient space Xˆ is Ti.

From the previous observations it follows the well known result that the quotient topology on ˆX isT0.

We now present some basic characterizations of the axiomsT_iρ, 1≤i≤4.

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(i) X isT₁ρ; the existence of an open neighborhood ofz,Wz such thatxρWz. But this contradicts

the fact thatz∈ {x}. Conversely, ifz∈¡T

Vx∈VxVx ¢

\{x}, thenzρx, from which there must exist a neighoborhoodUx ofxnot containingz, which is again a contradiction.

Conversely, considerxandy in X with xρy. Then eithery /∈ {x}or x /∈ {y}. In the first case, there is an open neighborhoodWy ofy, disjoint with{x}. It followsWyρx.

Assuming (ii) we obtain an open neighborhood Ux of x, not containing y, so that Uxρy. Te same argument applies to the casex /∈ {y}. Hence,X isT1ρ. ✷

Theorem 4.3. The following statements are equivalent:

(i) X isTR

It follows that z /∈ Wx, otherwise Wz and Wx would have non-empty intersection,

which contradicts (4.3).

On the other hand, if (ii) holds and xand y are elements in X such that xρy, then eitherx /∈ {y} ory /∈ {x}. Obviously, it is necessary to handle only one case, say the first. From (ii), we conclude that there exists a neighborhoodVy ofy such that

(4.4) x /∈Vy

and therefore, there exists an open set Vx ∈ Vx with Vx∩Vy = ¡❢. It is now clear

that

VxρVy.

This completes the proof. ✷

Lemma 4.4. The topological spaceX is regular if and only if it isρ-regular.

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Assuming now that x is ρ-regular, if x ∈ X and F is a closed subset of X with {x} ∩F = ¡❢, one can find a neighborhood Vx of xwith Vx∩F = ¡❢. Therefore, VxρF, from which one obtainsxρF. Letx∈UxandF⊆Ω whereVxand Ω are open

sets such thatUxρΩ. It is easy to see that Ux∩Ω =¡❢. The proof is now complete.

✷

Lemma 4.5. A topological spaceX is normal if and only if it isρ-normal.

Proof.The proof is an immediate consequence of the fact that any two open subsets ofX (respectively, any two closed subsets of X) are disjoint if and only if they are

ρ-disjoint. ✷

We are now in a position to prove the main Theorem of this Section, which is an immediate consequence of the above Lemmas:

Theorem 4.6. Let X be a topological space such that for each x∈X, the equality

(4.5) {x}= \

Vx∈Vx

Vx

holds. Let Xˆ =X/ρbe the quotient space, where ρ stands for Kolmogorov’s Second Equivalence Relation.

Then any two of the following statements are equivalent:

(i) Xˆ is aT1 space;

(ii) If X is regular, then Xˆ isT3;

(iii) If X is normal, then Xˆ isT4;

(iv) If{x}=T

Vx∈VxVx, then ˆ

X isT2.

✷

The following Remark shows that Theorem 4.6 provides a unifying framework for several otherwise isolated results in the theory of topological vector spaces and topo-logical groups.

Remark 4.7. (i) Condition (4.4) in Theorem4.6holds true in any topological group or topological vector space.

This is a consequence of the well known equality:

{0}= \

V∈V0

V,

whereV0stands for the filter of neighborhoods of the origin (see [4], Prop. 3.2, p. 32.)

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y(mod{0})s. This results from the fact that{0} is a normal subroup ofX ifX is a topological group or a closed subspace of X, if X is a topological vector space (see [B], Proposition 1, p. 226). Therefore,

xρy⇔ {x}={y} ⇔x+{0}=y+{0} ⇔x−y∈ {0}.

The previous Remark allows us to state the following:

Corollary 4.8. If X is a topological group, the quotient space Xˆ =X/{0} isT1. If X is a topological vector space, then Xˆ isT3.

Proof. The proof of the first assertion follows directly from part (i) of Theorem 4.6 and the previous Remark. The second proposition is a consequence of Theorem 8 (p.42) in [5] and (iii) in our Theorem 4.6. We underline the fact that Corollary 4.8 improves the corollary to Proposition 4.5 on page 34 of [4].

References

[1] N. Bourbaki,General Topology, Part I, Hermann 1966.

[2] J. Kelley,General Topology,Springer Verlag 1975.

[3] L. Popescu, R-separated spaces, Balkan Journal of Geometry and its Appl. 6 (2001), 81-88.

[4] F. Treves,Topological Vector Spaces, Distributions and Kernels, Academic Press 1967.

[5] A. Wilansky,Modern Methods in Topological Vector Spaces,Mc. Graw-Hill 1978.

Authors’ addresses:

Osvaldo Mendez

University of Texas at El Paso, Department of Math. Sciences, 500 W University Ave. El Paso, TX, 79968, USA.

E-mail: [email protected]

Liviu Horia Popescu

University of Oradea, Department of Mathematics, Str. Armatei Romane 5, 3700 Oradea, Romania. E-mail: [email protected]

Emil Daniel Schwab

University of Texas at El Paso, Department of Math. Sciences, 500 W University Ave. El Paso, TX, 79968, USA.