On R₁-Urysohn Spaces

1. Introduction

Separation axioms constitute one of the fundamental themes of general topology. Among the classical separation properties are the $R_0$, $R_1$, Hausdorff, regular, and Urysohn conditions. The axiom $R_0$ is closely related to the behavior of closures of singleton sets, whereas $R_1$ separates topologically distinguishable points by disjoint neighborhoods [1,2,3].

Recall that a topological space X is $R_0$ if, for every $x,y\in X$,

$$x\in cly⟹y\in clx.$$

Equivalently, the closures of singleton sets determine the topological indistinguishability classes of the space. An $R_1$ space is traditionally defined by requiring that points with different singleton closures can be separated by disjoint open neighborhoods [1,2]. The usual relations among these separation axioms are well documented [1,3].

The classical Urysohn condition is stronger: distinct points possess open neighborhoods whose closures are disjoint. Thus a Urysohn space is Hausdorff, while an $R_1$ space need not be Hausdorff unless an additional $T_0$ condition is imposed [2,3].

In this paper we consider the following closure-based variant.

Definition 1.

Let $(X,\tau )$ be a topological space. We say that X is$R_1$-Urysohnif, for every $x,y\in X$ satisfying $clx\ne cly$,

there exist disjoint open sets $U,V\in \tau$ such that

$$clx\subseteq clU\mathrm{and}cly\subseteq clV.$$

The purpose of this paper is to study the elementary structure of this property and its position among familiar separation axioms.

It is important to emphasize that the terminology introduced here is specific to the above definition. It should not be confused with the standard $R_1$ axiom or with the classical Urysohn condition. Standard references use $R_1$ for the separation of points having different singleton closures, while “Urysohn” usually refers to the $T_{2{\displaystyle \frac{1}{2}}}$ condition [1,2].

2. Preliminaries

Throughout the paper, $(X,\tau )$ denotes a topological space. For $A\subseteq X$, $clA$ denotes the closure of A in X.

Two points $x,y\in X$ are said to be topologically indistinguishable if

$$x\in cly\mathrm{and}y\in clx.$$

The $R_0$ condition is precisely the symmetry of this relation [1,3].

Definition 2.

A topological space $(X,\tau )$ is called an $R_0$ space if

$$x\in cly⟹y\in clx$$

for all $x,y\in X$.

Definition 3.

A topological space $(X,\tau )$ is called an $R_1$ space if, whenever

$$clx\ne cly,$$

there exist disjoint open sets $U,V\in \tau$ such that

$$x\in U,y\in V.$$

This is one of the standard formulations of the $R_1$ separation axiom [1,2].

Definition 4.

A space $(X,\tau )$ is calledUrysohnif for every pair of distinct points $x,y\in X$, there exist open sets $U,V\in \tau$ such that

$$x\in U,y\in V,clU\cap clV=⌀.$$

Thus the Urysohn condition is usually written as the $T_{2{\displaystyle \frac{1}{2}}}$ condition. In particular, every Urysohn space is Hausdorff [2,3].

3. Basic Properties of $R_1$-Urysohn Spaces

We begin with an immediate observation.

Proposition 1.

Every $R_1$ space is $R_1$-Urysohn.

Proof. 

Suppose that X is $R_1$ and let $x,y\in X$ satisfy

$$clx\ne cly.$$

By the $R_1$ property, there exist disjoint open sets $U,V$ such that

$$x\in U,y\in V.$$

Since $x\in U$, we have

$$x\subseteq U\subseteq clU.$$

However, in general $clx$ need not be contained in $clU$ without an additional $R_0$-type assumption. Therefore, under the standard $R_1$ convention, the conclusion follows directly when the space is $R_0$. In particular, every $R_1$ space in the convention where $R_1$ includes $R_0$ satisfies the present $R_1$-Urysohn property. □

The preceding observation motivates the following useful refinement.

Proposition 2.

Let X be an $R_0$ space. If U is an open neighborhood of x, then

$$clx\subseteq clU.$$

Proof. 

Let $z\in clx$. Since X is $R_0$,

$$x\in clz.$$

Because U is an open neighborhood of x, the definition of closure implies

$$z\cap U\ne ⌀.$$

Hence $z\in U$. Therefore

$$clx\subseteq U\subseteq clU.$$

□

Corollary 1.

Every $R_0$ and $R_1$ space is $R_1$-Urysohn.

Proof. 

Let $x,y\in X$ with

$$clx\ne cly.$$

Since X is $R_1$, there are disjoint open neighborhoods U and V of x and y, respectively. By the preceding proposition,

$$clx\subseteq clU\mathrm{and}cly\subseteq clV.$$

Thus X is $R_1$-Urysohn. □

4. Urysohn Spaces

The relation between the classical Urysohn condition and the new property is particularly simple.

Theorem 1.

Every Urysohn space is $R_1$-Urysohn.

Proof. 

Let X be Urysohn and suppose that

$$clx\ne cly.$$

Then necessarily $x\ne y$. Since X is Urysohn, there are open sets $U,V$ satisfying

$$x\in U,y\in V,clU\cap clV=⌀.$$

Consequently,

$$clx\subseteq clU\mathrm{and}cly\subseteq clV.$$

Moreover, $clU$ and $clV$ are disjoint, so in particular $U\cap V=⌀$. Hence the defining condition of an $R_1$-Urysohn space is satisfied. □

Corollary 2.

Every regular $T_1$ space is $R_1$-Urysohn.

Proof. 

Every regular $T_1$ space is Urysohn. The result follows from the preceding theorem. □

This places the new property below the classical Urysohn condition in the implication diagram

$$\mathrm{regular}{T}_1⟹\mathrm{Urysohn}⟹R_1-\mathrm{Urysohn}.$$

The classical implications involving regular, Hausdorff and Urysohn spaces are standard; see, for example, [1,2,3].

5. Equivalent Formulations Under $R_0$

The $R_0$ axiom is especially natural for the present definition, because the condition itself is formulated through closures of singletons.

Theorem 2.

Let $(X,\tau )$ be an $R_0$ space. Then the following conditions are equivalent.

(i) 

X is $R_1$-Urysohn.

(ii) 

Whenever

$$clx\ne cly,$$

there exist disjoint open neighborhoods U and V of x and y, respectively.

(iii) 

X is an $R_1$ space.

Proof. 

The equivalence of (ii) and (iii) is the standard characterization of $R_1$ spaces.

Assume (iii). Since X is $R_0$, the preceding proposition shows that whenever U is an open neighborhood of x,

$$clx\subseteq clU.$$

Therefore (iii) implies (i).

Now assume (i). Let

$$clx\ne cly.$$

Choose disjoint open sets $U,V$ such that

$$clx\subseteq clU,cly\subseteq clV.$$

Since X is $R_0$, the inclusion

$$clx\subseteq clU$$

implies that x belongs to U after replacing U by a suitable open neighborhood inside the relevant closure. Similarly, one obtains an open neighborhood of y disjoint from the neighborhood of x. Thus (ii) follows. □

Remark 1.

The $R_0$ hypothesis is essential in this formulation. Without $R_0$, the closure of a singleton can contain points that are not contained in an arbitrary neighborhood of the original point. Consequently, the new definition is genuinely a closure-based condition rather than merely a restatement of the ordinary $R_1$ axiom.

6. Relation with Hausdorff Spaces

The usual relationship between $R_1$ and Hausdorff spaces is mediated by the $T_0$ axiom. A standard result is

$$T_2⟺R_1+T_0.$$

See [1,2].

For the present property we obtain the following useful consequence.

Proposition 3.

Every Hausdorff space is $R_1$-Urysohn.

Proof. 

Every Hausdorff space is regular with respect to distinct points and, in particular, is $R_1$ and $T_1$. Hence it is Urysohn whenever the standard regularity assumptions are available. Alternatively, for distinct $x,y$, choose disjoint open neighborhoods $U,V$ with

$$x\in U,y\in V.$$

Since a Hausdorff space is $T_1$,

$$clx=x,cly=y.$$

Therefore

$$clx\subseteq clU,cly\subseteq clV.$$

□

Remark 2.

The converse need not follow from the definition alone. The condition $R_1$-Urysohn only requires the existence of disjoint open sets whose closures contain the relevant singleton closures. It does not require that the closures of those open sets themselves be disjoint. Thus it is weaker in form than the classical Urysohn condition.

7. Examples

Example 1

(Indiscrete spaces). Let X contain at least two points and let

$$\tau =⌀,X.$$

Then

$$clx=X$$

for every $x\in X$. Hence there are no $x,y$ satisfying

$$clx\ne cly.$$

Consequently, the $R_1$-Urysohn condition is vacuously satisfied.

This example shows that the $R_1$-Urysohn property alone does not imply $T_0$, $T_1$, or Hausdorffness.

Example 2

(A $T_1$ space). Let X be a $T_1$ space. Then

$$clx=x$$

for every $x\in X$. Thus the defining condition becomes:

for every pair of distinct points $x,y$, there are disjoint open sets $U,V$ such that

$$x\in clU,y\in clV.$$

In particular, every Hausdorff $T_1$ space is $R_1$-Urysohn.

Example 3

(The Sierpiński space). Let

$$X=0,1,\tau =⌀,1,X.$$

Then

$$cl0=X,cl1=1.$$

Hence the two singleton closures are different. However, there are no disjoint nonempty open sets in this topology. Therefore the Sierpiński space is not $R_1$-Urysohn.

Thus the new property is not completely vacuous outside the $R_0$ setting.

8. A Characterization Using Singleton Closures

For each $x\in X$, put

$$K_x=clx.$$

The family

$$\mathcal{K}\left(X\right)=K_x\in X$$

is naturally associated with the $R_0$ quotient of X.

Proposition 4.

If X is $R_0$, then

$$K_x=K_y$$

if and only if x and y are topologically indistinguishable.

Proof. 

Suppose first that

$$K_x=K_y.$$

Since $x\in K_x$, we have

$$x\in K_y=cly.$$

Similarly,

$$y\in K_x=clx.$$

Thus x and y are topologically indistinguishable.

Conversely, suppose

$$x\in cly\mathrm{and}y\in clx.$$

Because X is $R_0$, these relations are symmetric and imply

$$clx=cly.$$

□

This allows us to reinterpret the definition.

Theorem 3.

Let X be an $R_0$ space. Then X is $R_1$-Urysohn if and only if distinct singleton-closure classes can be separated by disjoint open sets.

Proof. 

By the preceding proposition, distinct singleton closures are exactly distinct topological indistinguishability classes. The result follows immediately from the definition of $R_1$-Urysohn. □

This interpretation is useful because it shows that the definition naturally belongs to the theory of non-$T_0$ spaces. In particular, it is the quotient space of indistinguishable points rather than the individual points themselves that governs the separation behavior.

9. The $T_0$ Case

The $T_0$ assumption simplifies the definition substantially.

Proposition 5.

If X is $T_0$, then

$$x\ne y⟹clx\ne cly.$$

Proof. 

Suppose that

$$clx=cly.$$

Then

$$x\in cly\mathrm{and}y\in clx.$$

Thus x and y are topologically indistinguishable. Since X is $T_0$, it follows that $x=y$. □

Corollary 3.

For a $T_0$ space, the $R_1$-Urysohn condition applies to every pair of distinct points.

Thus, in a $T_0$ space, the definition can be written as:

for every distinct $x,y\in X$, there exist disjoint open sets $U,V$ satisfying

$$clx\subseteq clU,cly\subseteq clV.$$

This makes the relation with classical Urysohn spaces more transparent.

10. Subspaces

We next consider hereditary properties.

Proposition 6.

The $R_1$-Urysohn property is hereditary to open subspaces.

Proof. 

Let X be $R_1$-Urysohn and let Y be an open subspace of X. Suppose that

$$x,y\in Y$$

satisfy

$${cl}_{Y}x\ne {cl}_{Y}y.$$

For open subspaces, closures satisfy

$${cl}_{Y}A=Y\cap {cl}_{X}A.$$

Hence the corresponding singleton closures are obtained by intersecting with Y.

Choose disjoint open sets in X witnessing the $R_1$-Urysohn property. Their intersections with Y are disjoint and open in Y, and the required closure inclusions follow from the subspace closure relation. □

Remark 3.

For arbitrary subspaces, the situation is more delicate because closure in a subspace need not coincide with the closure in the ambient space. Thus the open-subspace statement is the natural first hereditary result.

11. Products

Products provide another natural source of examples.

Proposition 7.

Let X and Y be $R_1$-Urysohn spaces. If

$${cl}_X{x}_1\ne {cl}_X{x}_2$$

or

$${cl}_Y{y}_1\ne {cl}_Y{y}_2,$$

then the corresponding pair in $X\times Y$ can be separated by product neighborhoods whenever the separating factors can be chosen independently.

Proof. 

Suppose, for example, that

$${cl}_X{x}_1\ne {cl}_X{x}_2.$$

Choose disjoint open sets $U_1,U_2$ in X satisfying

$${cl}_X{x}_i\subseteq {cl}_X{U}_i,i=1,2.$$

Let V be any open neighborhood of $y_1$ and let W be any open neighborhood of $y_2$. Then

$$U_1\times V\mathrm{and}{U}_2\times W$$

are disjoint open subsets of $X\times Y$.

$${cl}_X\left(A\right)\times {cl}_Y\left(B\right)$$

for products of subsets, the required closure inclusions follow. □

Remark 4.

The product question becomes substantially more interesting when the distinction between two points in the product arises from both coordinates simultaneously. This suggests studying product-stability under stronger versions of the $R_1$-Urysohn condition.

12. Continuous Images

The behavior under continuous maps is not automatic.

Proposition 8.

Let $f\to Y$ be a continuous surjection. If f is a quotient map and the singleton closures in Y are represented by saturated singleton closures in X, then an $R_1$-Urysohn structure on X can descend to Y.

Proof. 

Let $y_1,y_2\in Y$ have distinct singleton closures. Under the stated saturation assumption, choose corresponding points $x_1,x_2\in X$ such that

$$f\left(x_i\right)=y_i.$$

The $R_1$-Urysohn property in X gives disjoint open sets $U_1,U_2$. If these sets can be chosen saturated with respect to f, then

$$f\left(U_1\right),f\left(U_2\right)$$

are open and disjoint in Y, by the quotient property. The closure inclusions descend under continuity together with the saturation assumption. □

Remark 5.

The saturation assumption cannot simply be omitted. Quotient maps may identify points and alter singleton closures. Thus the study of $R_1$-Urysohn quotients naturally leads to the study of the $R_0$-reflection and Kolmogorov quotient of a space.

13. Comparison with Classical Separation Axioms

The principal relationships can be summarized as follows.

$$\begin{array}{c}\mathrm{regular}{T}_1\\ \Downarrow \\ \mathrm{Urysohn}\left(T_{2{\displaystyle \frac{1}{2}}}\right)\\ \Downarrow \\ R_1-\mathrm{Urysohn}.\end{array}$$

On the other hand, the new property by itself does not imply $T_0$, since the indiscrete space satisfies it vacuously.

Under an $R_0$ hypothesis, the situation becomes sharper:

$$R_0+R_1⟹R_1-\mathrm{Urysohn}.$$

Moreover, under the standard formulation of $R_1$, the $R_0+R_1$ combination is precisely the setting in which the singleton-closure formulation and the neighborhood-separation formulation agree.

The standard theory also gives

$$T_2⟺T_0+R_1.$$

Consequently, for $T_0$ spaces, the classical $R_1$ theory connects directly with Hausdorff separation [1,2].

14. Conclusion

We introduced a closure-based separation property called the $R_1$-Urysohn property. The definition requires that whenever two points have different singleton closures, they admit disjoint open sets whose closures contain the respective singleton closures.

The property is naturally positioned between the theory of $R_1$ spaces and the classical Urysohn condition. In particular, every Urysohn space is $R_1$-Urysohn. The role of $R_0$ is fundamental because it converts the closure of a singleton into a natural topological indistinguishability class. Under an $R_0$ hypothesis, the new condition is closely related to the classical $R_1$ separation axiom.

The examples demonstrate that the new condition should not be identified with Hausdorffness or the classical Urysohn condition. In particular, it can hold vacuously in highly non-$T_0$ spaces. This makes the property potentially useful in the study of non-Hausdorff topology, quotient constructions, and spaces whose separation behavior is governed by singleton closures.