#### Abstract

In 1972, Bennett studied the countable dense homogeneous (CDH) spaces and in 1992, Fitzpatrick, White, and Zhou proved that every CDH space is a space. Afterward Bsoul, Fora, and Tallafha gave another proof for the same result, also they defined the almost CDH spaces and almost , spaces, indeed they prove that every ACDH space is an almost space. In this paper we introduce a new type of almost CDH spaces called ACDH-1, we characterize the ACDH spaces, the almost spaces, we also give relations between different types of CDH spaces. We define new type of almost () spaces, and we study the relations between the old and new definitions. By extending the techniques given by Tallafha, Bsoul, and Fora, we prove that every ACDH-1 is an .

#### 1. Introduction

In 1920, Sierpinski introduced in [1] the notion of homogeneous spaces, saying that a topological space is a homogeneous space if for any in , then there is a homeomorphism of such that . Frรฉchet in [2] and Brouwer in [3] observed that the -dimensional Euclidean space has the property that if are countable dense subsets of , then there is a homeomorphism of such that . Afterward, in 1972 the abstract study was begun by Bennett in [4], who called such spaces the countable dense homogeneous (CDH) spaces.

In 1974, Lauer defined in [5] the densely homogeneous (DH) spaces, and in 1992, Fitzpatrick et al. proved in [6] that DH and CDH spaces are spaces, and afterward Tallafha et al. in [7] gave another proof for the fact that CDH spaces are spaces. In [8], Fora et al. defined almost CDH (ACDH) spaces, almost () spaces, and they discussed the relation between such spaces.

In the first part of this paper, we introduce the definitions given by Fora et al. in [8] for almost CDH spaces, almost spaces. We give a characterization of almost CDH spaces besides we define a new type of almost CDH (ACDH-1) spaces, Transposition Homogeneous space (TH). Also, we discuss the relation between the new type and the others.

In Section 4 we give a characterization of almost we introduce new definition of almost space and we study the relation between them and the old ones. We finally prove our main result using the idea of almost closurely ordered sets, more precisely, we show that every almost CDH space of type 1 (ACDH-1) is an space.

Finally, the following abbreviations and symbols will be used throughout this paper. For a subset of a topological space , we write , or for the closure of , for , by which denotes the closure of and which denotes the cardinality of . By we mean the set of all homeomorphisms of , refers to the symmetric difference of sets, , , and denote the set with the indiscrete, discrete, cofinite, and cocountable topologies, and , and denote with the left ray, right ray, and the usual topologies, respectively.

#### 2. Almost CDH Spaces

Fora et al. in [8] defined almost and strong almost countable dense homogeneous spaces.

*Definition 2.1 (see [8]). *A space is called an almost countable dense homogeneous (ACDH) space if it is a separable space and for any two countable dense subsets , there are two finite subsets , and such that and . In addition, if , then is called a strong ACDH space which is denoted by SACDH.

*Remark 2.2. *In the previous definition the condition is redundant since if there exist finite sets and such that and , then we choose the finite sets and as follows , where .

Theorem 2.3 (see [8]). *If X is countable and is SACDH, then is the discrete topology.*

Let be any topological space. For , let , , , , and . And Let .

*Definition 2.4 (see [8]). *A topological space is called almost ( ) if . If, in addition, , then is called strong almost ().

Clearly every space is .

*Definition 2.5 (see [8]). *A topological space is called almost and denoted by if it is and , that is, are all finite sets.

*Definition 2.6 (see [8]). *A topological space is called strong almost space () if . Clearly every space is and every is

Theorem 2.7 (see [8]). *Every ACDH space is .*

Theorem 2.8 (see [8]). *Let be ACDH space. If , then is .*

Now let us define a new type of almost CDH spaces.

*Definition 2.9. *A topological space is called an almost CDH of type 1 (ACDH-1) if it is separable space and there exists a finite subset such that, for any two countable dense subsets and of , there exists such that . Note that from now on, we will refer to F by a related finite set.

Clearly every finite space is an almost CDH of type 1. Moreover, we have the following result.

Proposition 2.10. *If is ACDH-1, , and F is a related finite set, then for all , we have which is an open set in X.*

*Proof. *Let . If is not an open set in , then , therefore there is such that which is a contradiction.

As an application of Proposition 2.10, we have the following corollary.

Corollary 2.11. *If is an ACDH-1, , then there exists the smallest related finite set .*

*Proof. *Let be a nondiscrete ACDH-1 space and not open. As , Also by Proposition 2.10, , for all related finite sets , so is finite. If are two dense sets and , then and hence therefore is the smallest finite related set. Moreover, if , then .

The following example shows that ACDH-1 space need be a CDH space.

*Example 2.12. *Let and . Let be two dense subsets of . Then let , hence is ACDH-1 space which is not a CDH space as it is not a space.

Now let prove the following characterizations of ACDH spaces.

Theorem 2.13. *If is a separable space, then the following are equivalent *(i)* is an ACDH space*(ii)*For any two countable dense subsets , there exist two finite subsets of , respectively, and such that .*(iii)*For any two countable dense subsets , there exist two equipotent finite subsets and such that .*(iv)*For any two countable dense subsets , there exist two finite subsets and such that .*

*Proof. *(i) implies (ii) Suppose that is ACDH space. Let be two countable dense sets. Then there exist two finite sets and such that and . Let , so are two finite sets and
(ii) implies (iii) Let be two countable dense sets, then there exist two finite sets , and such that , and . Let and . Clearly . Moreover,
But clearly , so . (iii) implies that (iv) is clear.

(iv) implies (i) Let be two countable dense sets, then there are finite sets and such that .

Let and . Then are two finite sets. If , then and which gives a contradiction. Similarly, . Moreover,

Consequently, we have the following result.

Corollary 2.14. *Every ACDH-1 space is an ACDH space.*

Theorem 2.15. *Let be a separable space. Then is an SACDH space if and only if for every two countable dense sets , there exist two equipotent finite subsets of respectively, and such that .*

*Proof. *Let be two countable dense sets, so there exist , , , and such that . Let and . It is clear that . Also,
Moreover,
Hence, is an SACDH space.

Conversely, assume that is an SACDH space. Let be two countable dense sets, therefore, there are two finite sets with and such that ; moreover, . Let and . Then , moreover, and . We need to prove that Now, . Claim that . If , then . As , , so . To prove the other inclusion, suppose that . Therefore, , for some , and then so . Also , hence .

Consequently, we have the following result.

Corollary 2.16. *Every ACDH-1 space is SACDH.*

#### 3. T-Homogeneous Spaces

A transposition on is a permutation on which exchanges the places of two elements while leaving all the other elements unchanged. Now we will define the Transposition-Homogeneous (TH) spaces, and we will show that every TH SACDH space is a CDH space.

*Definition 3.1. *A space is called Transposition-Homogeneous (TH) space if every transposition on is a homeomorphism.

Proposition 3.2. *A space is (TH) if and only if, for any two finite subsets with the same cardinality and for every , there exists such that *(1)*,*(2)*, for every ,*(3)*.*

*Proof. *Let be a TH space and a finite subset of Let be a permutation which fixes clearly is a composition of finite transpositions which is a homeomorphism. Let be two finite subsets with the same cardinality and clearly and are a disjoint union of finite sets which is finite. Let be a bijection such that For each let be a transposition on which transposes and Now let be the finite composition of the transpositions Then for all , , and Now is the required function since , for every

The converse is obvious by choosing to be the identity and ,

*Example 3.3. *One can show that the spaces , , , and are all TH spaces. However, the spaces and are not TH spaces.

The following example shows that ACDH-1, TH space need be CDH space, hence ACDH TH space need be a CDH space.

*Example 3.4. *Let be such that , with the indiscrete topology. The space is a TH-space and it is also ACDH-1, but it is not a CDH space as it is not a space.

Theorem 3.5. *If is SACDH, TH space, then is a CDH space.*

*Proof. *If , are two countable dense subsets of , then there exist two finite subsets of , respectively with and there is such that . As is a TH space, there exists such that , , for all and . To show that we show first that . Suppose that there is Then which gives a contradiction. Hence , so .

Let be an ACDH-1 space and and define , and also we define the relation on by: if and only if there is such that . Let

Now we have the following result.

Theorem 3.6. *If is an ACDH-1, TH space, and , then there is such that .*

*Proof. *Suppose that , so there are and such that with . If , then we are done. In general let and , therefore . As is a TH space, there exists such that , for all and , also . Now , so .

Consequently, we have the following Corollary.

Corollary 3.7. *If is an ACDH-1, TH space, and , then is a CDH space.*

#### 4. Almost CDH Spaces, and New Separation Axioms

We know that almost CDH space is not a space. In this section we will give a characterization of almost space, also we will give a new definition of almost space.

Theorem 4.1. *Let be a topological space. Then is space if and only if there exists a finite subset of , such that, for all and , there is an open set containing only one of , . We will refer to by a related finite set .*

*Proof. *Assume that is an space. Let . Then for all with , we have that and , hence or Conversely, suppose that there exists a finite set such that for all with , there is an open set containing only one of , . If is an infinite, then there exists a denumerable subset of , say , and , for all , so there exist and for all . Therefore there is such that are both not in and , which gives a contradiction.

*Definition 4.2. *A space is called an almost strong () space if there is a finite subset of such that, for all and , there are two open subsets of , such that and . is called the related finite set.

One may easily prove the following proposition.

Proposition 4.3. *Let be a topological space. If is an space, then for all , we have ; where is a related finite set. Conversely, if there is a finite set such that for all , , then is an .*

In the following results we show that the new separation axiom is stronger than the one defined by Fora et al. in [8].

Proposition 4.4. *Every space is space.*

*Proof. *Suppose that is an space, and let be a related finite set. By Theorem 4.1 it is an , therefore, . By Proposition 4.3, for all , we have so that . If , then there is such that for all , and , for all . Let be such that . Therefore , then there is , so , which gives a contradiction, and hence the proposition is proved.

The following example shows that the converse of the previous proposition need not be true.

*Example 4.5. *Let so is a base for some topology on . Note that for , we have

Therefore, for all , , hence , as for all , . Now for
Therefore, , and then is space. Let be any finite subset of . Let , as and are both not in , is not an space.

One may easily prove the following proposition.

Proposition 4.6. *Every space is space.*

Fitzpatrick et al. proved in [6] that every CDH space is a space. Indeed, Tallafha et al. in [7] gave us another proof for the same argument by using the idea of closurely ordered sets. Now, we will prove that every ACDH-1 space is space by using the idea of almost closurely ordered sets.

*Definition 4.7 (see [7]). *Let be a topological space. A countable subset of is said to have the closurely ordered property if there exists a numeration of say such that for all , . The numeration is called closurely ordered countable set.

*Definition 4.8 (see [7]). *A countable collection of subsets of is said to have the closurely ordered countable property if can be written as , where . The form is called closurely ordered countable family.

Theorem 4.9 (see [7]). *Let be a topological space and let be any countable dense subset of . Then there exists a countable dense subset of , such that is closurely ordered countable set.*

Theorem 4.10 (see [7]). *Let be a topological space, then, *(i)*if is an injective open function and has the closurely ordered property in , then has the closurely ordered property in ,*(ii)*having closurely ordered property, is a topological property,*(iii)*every subset of a set having closurely ordered property must have closurely ordered property.*

Now let us define the following.

*Definition 4.11. *A countable set in is said to have the almost closurely ordered property if there is a finite set in such that has the closurely ordered property. If is a closurely ordered set, then is called almost closurely ordered set.

Proposition 4.12. *If is an space and F is a related finite set, then each doubleton has the closurely ordered property. Conversely, in a topological space if there exists a finite set all doubletons have the closurely ordered property, then , is an space.*

*Proof. *The first part is clear. To prove the converse, assume that there is such a finite set . Let be such that and . So or by Theorem 4.1 () is an

Theorem 4.13. *Every ACDH-1 space is space.*

*Proof. *Assume that is an ACDH-1 space and is a related finite set. We want to show that is the desired set. If with and is a countable dense subset of , by Theorem 4.9, we may assume that has the closurely ordered property, also by Theorem 4.10, has the closurely ordered property. Now is also a countable dense subset of , therefore there is such that . So has the closurely ordered property, the result follows by Theorem 4.10.

Theorem 4.14 (see [7]). *If a topological space and K is a countable dense subset, then there exists a countable collection of subsets of such that *(i)*,*(ii)* is closurely ordered countable family,*(iii)*each has the closurely ordered property,*(iv)*each is either a singleton or an infinite set,*(v)*,*(vi)*if is a singleton, say , then and , for all ,*(vii)*if is infinite set, then , for all .*

It is easy to prove the following result.

Proposition 4.15. *The properties (i)โ(vii) in the last theorem are all preserved under homeomorphisms.*

We now prove the following theorem, that will be used to prove our main result.

Theorem 4.16. *Let be an ACDH-1 space and let be a related finite set. If and , then is an infinite set.*

*Proof. *Suppose that and . Suppose that is a finite set, then there is such that . Let be any countable dense set in , then by Theorem 4.14, there is a countable collection of subsets of say satisfing the conditions (i)โ(vii). So, and . Let . For , define and for , also define . To show that . If , then , therefore . Then is a countable dense set in and so is . Therefore, there exists such that . As , we have for some . If , then and , so where ; therefore, which is a contradiction by Theorem 4.14(vi). If and for some , so all are in . As , which is impossible, so is an infinite set.

Recall that in ACDH-1 space, if is a countable dense subset of , then by Theorem 4.14 there are countable subsets of satisfying (i)โ(vii) of the pointed theorem.

Moreover, , . Therefore, there is such that . By Proposition 4.15, can be decomposed in the same way as

The following theorem shows that all the above s are singletons.

Theorem 4.17. *Let be an ACDH-1 space, K any countable dense subset of , and a related finite set. Then and .*

*Proof. *Let . If , then , for some . We have and , so, by Theorem 4.16, we have which is an infinite set. Let with . In a similar way, let with . By the same argument, we have a sequence and . Now we claim that for all . If for some and , then which contradicts the fact that is a closurely ordered set. Also for all , since and , so we proved our claim. Let . Then , hence is a countable dense subset of . Then there is such that . For and , for all . Then by Proposition 4.15, , . Now define , therefore . Moreover, , for some , where . Then , for some . Also , where and . If , for some , then which is a contradiction, so

As a consequence of the previous theorem, we have the following results.

Corollary 4.18. *If is an ACDH-1 space, is a related finite set, and is a countable dense subset of , then has the closurely ordered property.*

*Proof. *If is a countable dense subset of , then by using Theorem 4.17, we have that and , for all . Therefore, indeed, , for all .

Corollary 4.19. *Every ACDH-1 space is an space.*

*Proof. *Let be a related finite set and with . If is a countable dense subset of , say , then the set is also a countable dense subset of , therefore by Corollary 4.18, we have that has the closurely ordered property and , therefore . Similarly, .

#### Acknowledgment

This paper is financially supported by the Deanship of Academic Research at the University of Jordan, Amman, Jordan.