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 ๐‘‡1 space. Afterward Bsoul, Fora, and Tallafha gave another proof for the same result, also they defined the almost CDH spaces and almost ๐‘‡1, ๐‘‡0 spaces, indeed they prove that every ACDH space is an almost ๐‘‡1 space. In this paper we introduce a new type of almost CDH spaces called ACDH-1, we characterize the ACDH spaces, the almost ๐‘‡0 spaces, we also give relations between different types of CDH spaces. We define new type of almost ๐‘‡1 (๐ด๐‘‡1) 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 ๐ด๐‘‡1.

1. Introduction

In 1920, Sierpinski introduced in [1] the notion of homogeneous spaces, saying that a topological space ๐‘‹ is a homogeneous space if for any ๐‘ฅโ‰ ๐‘ฆ in ๐‘‹, then there is a homeomorphism โ„Ž of ๐‘‹ such that โ„Ž(๐‘ฅ)=๐‘ฆ. Frรฉchet in [2] and Brouwer in [3] observed that the ๐‘›-dimensional Euclidean space โ„๐‘› has the property that if ๐ด,๐ต are countable dense subsets of โ„๐‘›, then there is a homeomorphism โ„Ž of โ„๐‘› such that โ„Ž(๐ด)=๐ต. 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 ๐‘‡1 spaces, and afterward Tallafha et al. in [7] gave another proof for the fact that CDH spaces are ๐‘‡1 spaces. In [8], Fora et al. defined almost CDH (ACDH) spaces, almost ๐‘‡0,๐‘‡1 (๐ด๐‘‡0,๐ด๐‘‡1) 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 ๐‘‡0,๐‘‡1 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 ๐‘‡0 we introduce new definition of almost ๐‘‡1 space ๐ด๐‘†๐‘‡1, 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 ๐ด๐‘†๐‘‡1 space.

Finally, the following abbreviations and symbols will be used throughout this paper. For a subset ๐ด of a topological space (๐‘‹,๐œ), we write ๐ด, or Cl(๐ด) for the closure of ๐ด, for ๐‘ฅโˆˆ๐‘‹, by ๐‘ฅ which denotes the closure of {๐‘ฅ} and |๐ด| which denotes the cardinality of ๐ด. By ๐ป(๐‘‹) we mean the set of all homeomorphisms of ๐‘‹, ฮ” refers to the symmetric difference of sets, (๐‘‹,๐œind), (๐‘‹,๐œdis), (๐‘‹,๐œcof), and (๐‘‹,๐œcoc) denote the set ๐‘‹ with the indiscrete, discrete, cofinite, and cocountable topologies, and (โ„,๐œ๐‘™โ‹…๐‘Ÿ),(โ„,๐œ๐‘Ÿโ‹…๐‘Ÿ), and (โ„,๐œ๐‘ข) denote โ„, 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 (๐‘‹,๐œ) is called an almost countable dense homogeneous (ACDH) space if it is a separable space and for any two countable dense subsets ๐พ1,๐พ2, there are two finite subsets ๐น1,๐น2, ๐พ1โˆฉ๐น1=๐พ2โˆฉ๐น2=โˆ… and โ„Žโˆˆ๐ป(๐‘‹) such that โ„Ž(๐พ1โˆช๐น1)=๐พ2โˆช๐น2 and โ„Ž(๐น1)โˆฉ๐น2=โˆ…. In addition, if |๐น1|=|๐น2|, then (๐‘‹,๐œ) is called a strong ACDH space which is denoted by SACDH.

Remark 2.2. In the previous definition the condition โ„Ž(๐น1)โˆฉ๐น2=โˆ… is redundant since if there exist finite sets ๐ต1 and ๐ต2 such that ๐พ1โˆฉ๐ต1=๐พ2โˆฉ๐ต2=โˆ… and โ„Ž(๐พ1โˆช๐ต1)=๐พ2โˆช๐ต2, then we choose the finite sets ๐น1 and ๐น2 as follows ๐น1=๐ต1โงตโ„Žโˆ’1(๐ถ),๐น2=๐ต2โงต๐ถ, where ๐ถ=โ„Ž(๐ต1)โˆฉ๐ต2.

Theorem 2.3 (see [8]). If X is countable and (๐‘‹,๐œ) is SACDH, then ๐œ is the discrete topology.

Let (๐‘‹,๐œ) be any topological space. For ๐‘ฅโˆˆ๐‘‹, let ๐ด๐‘ฅ={๐‘ฆโˆˆ๐‘‹โˆถ๐‘ฆ=๐‘ฅ}, ๐ด={๐‘ฅโˆถ|๐ด๐‘ฅ|>1}, ๐ต={๐‘ฅโˆถ|๐‘ฅ|โ‰ฅโ„ต0}, ๐ถ๐‘ฅ={๐‘ฆโˆˆ๐‘‹โˆถ๐‘ฅโˆˆ๐‘ฆ}, and ๐ถ={๐‘ฅโˆถ|๐ถ๐‘ฅ|โ‰ฅโ„ต0}. And Let ๐น={๐‘ฅโˆˆ๐‘‹โˆถ{๐‘ฅ}isnotaclosedset}.

Definition 2.4 (see [8]). A topological space (๐‘‹,๐œ) is called almost ๐‘‡๐‘œ (๐ด๐‘‡0 ) if |๐ด|<โ„ต0. If, in addition, |๐ต|<โ„ต0, then (๐‘‹,๐œ) is called strong almost ๐‘‡๐‘œ (๐‘†๐ด๐‘‡๐‘œ).

Clearly every ๐‘‡๐‘œ-space is ๐ด๐‘‡๐‘œ.

Definition 2.5 (see [8]). A topological space (๐‘‹,๐œ) is called almost ๐‘‡1 and denoted by ๐ด๐‘‡1 if it is ๐‘†๐ด๐‘‡๐‘œ and |๐ถ|<โ„ต0, that is, ๐ด,๐ต,and๐ถ are all finite sets.

Definition 2.6 (see [8]). A topological space (๐‘‹,๐œ) is called strong almost ๐‘‡1 space (๐‘†๐ด๐‘‡1) if |๐น|<โ„ต๐‘œ. Clearly every ๐‘‡1 space is ๐‘†๐ด๐‘‡1 and every ๐‘†๐ด๐‘‡1 is ๐ด๐‘‡1.

Theorem 2.7 (see [8]). Every ACDH space is ๐ด๐‘‡0.

Theorem 2.8 (see [8]). Let (๐‘‹,๐œ) be ACDH space. If |๐‘‹|โ‰คโ„ต0, then (๐‘‹,๐œ) is ๐ด๐‘‡1.

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

Definition 2.9. A topological space (๐‘‹,๐œ) is called an almost CDH of type 1 (ACDH-1) if it is separable space and there exists a finite subset ๐น such that, for any two countable dense subsets ๐ด and ๐ต of ๐‘‹, there exists โ„Žโˆˆ๐ป(๐‘‹) such that โ„Ž(๐ดโงต๐น)=๐ตโงต๐น. 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 (๐‘‹,๐œ) is ACDH-1, |๐‘‹|โ‰คโ„ต0, and F is a related finite set, then for all ๐‘ฅโˆ‰๐น, we have {๐‘ฅ} which is an open set in X.

Proof. Let ๐‘ฅโˆˆ๐‘‹โงต๐น. If {๐‘ฅ} is not an open set in ๐‘‹, then ๐‘‹โงต{๐‘ฅ}=๐‘‹, therefore there is โ„Žโˆˆ๐ป(๐‘‹) such that โ„Ž(๐‘‹โงต๐น)=(๐‘‹โงต{๐‘ฅ})โงต๐น which is a contradiction.

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

Corollary 2.11. If (๐‘‹,๐œ) is an ACDH-1, |๐‘‹|โ‰คโ„ต0, then there exists the smallest related finite set ๐ด.

Proof. Let (๐‘‹,๐œ) be a nondiscrete ACDH-1 space and ๐ด={๐‘ฅโˆˆ๐‘‹โˆถ{๐‘ฅ} not open}. As ๐œโ‰ ๐œdis, ๐ดโ‰ ๐œ™. Also by Proposition 2.10, ๐ดโІ๐น, for all related finite sets ๐น, so ๐ด is finite. If ๐พ1,๐พ2 are two dense sets and โ„Žโˆˆ๐ป(๐‘‹), then ๐‘‹โงต๐ดโІ๐พ1โˆฉ๐พ2 and โ„Ž(๐‘‹โงต๐ด)=๐‘‹โงต๐ด, hence โ„Ž(๐พ1โงต๐ด)=โ„Ž(๐‘‹โงต๐ด)=๐‘‹โงต๐ด=๐พ2โงต๐ด; therefore ๐ด is the smallest finite related set. Moreover, if ๐œ=๐œdis, then ๐ด=โˆ….

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

Example 2.12. Let ๐‘‹=๐ and ๐›ฝ={{1,2},{3,4},{5},{6},โ€ฆ}. Let ๐ด,๐ต be two dense subsets of ๐‘‹. Then {5,6,7,โ€ฆ}โІ๐ดโˆฉ๐ต let ๐น={1,2,3,4}, hence (๐‘‹,๐œ(๐›ฝ)) is ACDH-1 space which is not a CDH space as it is not a ๐‘‡1 space.

Now let prove the following characterizations of ACDH spaces.

Theorem 2.13. If (๐‘‹,๐œ) is a separable space, then the following are equivalent (i)(๐‘‹,๐œ) is an ACDH space(ii)For any two countable dense subsets ๐พ1,๐พ2, there exist two finite subsets ๐น1,๐น2 of ๐พ1,๐พ2, respectively, and โ„Žโˆˆ๐ป(๐‘‹) such that โ„Ž(๐พ1โงต๐น1)=๐พ2โงต๐น2.(iii)For any two countable dense subsets ๐พ1,๐พ2, there exist two equipotent finite subsets ๐น1,๐น2, and โ„Žโˆˆ๐ป(๐‘‹) such that โ„Ž(๐พ1โงต๐น1)=๐พ2โงต๐น2.(iv)For any two countable dense subsets ๐พ1,๐พ2, there exist two finite subsets ๐น1,๐น2, and โ„Žโˆˆ๐ป(๐‘‹) such that โ„Ž(๐พ1โงต๐น1)=๐พ2โงต๐น2.

Proof. (i) implies (ii) Suppose that (๐‘‹,๐œ) is ACDH space. Let ๐พ1,๐พ2 be two countable dense sets. Then there exist two finite sets ๐บ1,๐บ2 and โ„Žโˆˆ๐ป(๐‘‹) such that ๐บ1โˆฉ๐พ1=๐บ2โˆฉ๐พ2=๐บ2โˆฉโ„Ž(๐บ1)=โˆ… and โ„Ž(๐พ1โˆช๐บ1)=๐พ2โˆช๐บ2. Let ๐น1=โ„Žโˆ’1(๐บ2)and๐น2=โ„Ž(๐บ1), so ๐น1,๐น2 are two finite sets and โ„Ž๎€ท๐พ1โงต๐น1๎€ธ=๎€ทโ„Ž๎€ท๐พ1โˆช๐บ1๎€ธ๎€ท๐บโงตโ„Ž1๎€ธ๎€ธโงต๐บ2=๐พ2โงต๐น2.(2.1) (ii) implies (iii) Let ๐พ1,๐พ2 be two countable dense sets, then there exist two finite sets ๐น1,๐น2, and โ„Žโˆˆ๐ป(๐‘‹) such that ๐น1โІ๐พ1,๐น2โІ๐พ2, and โ„Ž(๐พ1โงต๐น1)=๐พ2โงต๐น2. Let ๐บ1=๐น1โˆชโ„Žโˆ’1(๐น2) and ๐บ2=โ„Ž(๐น1)โˆช๐น2. Clearly |๐บ1|=|๐บ2|. Moreover, โ„Ž๎€ท๐พ1โงต๐บ1๎€ธ๎€ท๐พ=โ„Ž1โงต๐น1โˆชโ„Žโˆ’1๎€ท๐น2๎€ธ๎€ธ=๐พ2โงต๐น2.(2.2) But clearly (โ„Ž(๐น1)โงต๐น2)โˆฉ๐พ2=โˆ…, so ๐พ2โงต๐น2=๐พ2โงต๐บ2. (iii) implies that (iv) is clear.
(iv) implies (i) Let ๐พ1,๐พ2 be two countable dense sets, then there are ๐บ1,๐บ2 finite sets and โ„Žโˆˆ๐ป(๐‘‹) such that โ„Ž(๐พ1โงต๐บ1)=๐พ2โงต๐บ2.
Let ๐น1=โ„Žโˆ’1(๐บ2โˆฉ๐พ2)โงต(๐บ1โˆฉ๐พ1) and ๐น2=โ„Ž(๐บ1โˆฉ๐พ1)โงต(๐บ2โˆฉ๐พ2). Then ๐น1,๐น2 are two finite sets. If ๐‘กโˆˆ๐น1โˆฉ๐พ1, then ๐‘กโˆˆ๐พ1โงต๐บ1 and โ„Ž(๐‘ก)โˆˆ๐บ2โˆฉ๐พ2 which gives a contradiction. Similarly, ๐น2โˆฉ๐พ2=โˆ…. Moreover, โ„Ž๎€ท๐พ1โˆช๐น1๎€ธ๎€ท๐พ=โ„Ž1๎€ธโˆช๐บ๎€บ๎€ท2โˆฉ๐พ2๎€ธ๎€ท๐บโงตโ„Ž1โˆฉ๐พ1๎€ท๐พ๎€ธ๎€ป=โ„Ž1๎€ธโˆช๎€ท๐บ2โˆฉ๐พ2๎€ธ๐พ=โ„Ž๎€ท๎€ท1โงต๐บ1๎€ธโˆช๎€ท๐พ1โˆฉ๐บ1โˆช๎€ท๐บ๎€ธ๎€ธ2โˆฉ๐พ2๎€ธ=๐พ2โงต๐บ2๎€ท๐พโˆชโ„Ž1โˆฉ๐บ1๎€ธโˆช๎€ท๐บ2โˆฉ๐พ2๎€ธ=๐พ2โˆช๎€ทโ„Ž๎€ท๐พ1โˆฉ๐บ1๎€ธโงต๐พ2๎€ธ=๐พ2โˆช๐น2.(2.3)

Consequently, we have the following result.

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

Theorem 2.15. Let (๐‘‹,๐œ) be a separable space. Then (๐‘‹,๐œ) is an SACDH space if and only if for every two countable dense sets ๐พ1,๐พ2, there exist two equipotent finite subsets ๐น1,๐น2 of ๐พ1,๐พ2, respectively, and โ„Žโˆˆ๐ป(๐‘‹) such that โ„Ž(๐พ1โงต๐น1)=๐พ2โงต๐น2.

Proof. Let ๐พ1,๐พ2 be two countable dense sets, so there exist โ„Žโˆˆ๐ป(๐‘‹), ๐บ1โІ๐พ1, ๐บ2โІ๐พ2, and |๐บ1|=|๐บ2|<โ„ต0 such that โ„Ž(๐พ1โงต๐บ1)=๐พ2โงต๐บ2. Let ๐น1=โ„Žโˆ’1(๐บ2)โงต๐พ1 and ๐น2=โ„Ž(๐บ1)โงต๐พ2. It is clear that ๐น1โˆฉ๐พ1=๐น2โˆฉ๐พ2=โˆ…. Also, ||๐น1||=||โ„Žโˆ’1๎€ท๐บ2๎€ธโงต๐พ1||=||โ„Žโˆ’1๎€ท๐บ2๎€ธโงต๎€ท๐พ1โˆฉโ„Žโˆ’1๎€ท๐บ2||=||โ„Ž๎€ธ๎€ธโˆ’1๎€ท๐บ2๎€ธ||โˆ’||๐พ1โˆฉโ„Žโˆ’1๎€ท๐บ2๎€ธ||=||๐บ1||โˆ’||๎€ทโ„Ž๎€ท๐พ1โงต๐บ1๎€ธ๎€ท๐บโˆชโ„Ž1๎€ธ๎€ธโˆฉ๐บ2||=||๐บ1||โˆ’||โ„Ž๎€ท๐บ1๎€ธโˆฉ๐บ2||=||๐บ1||โˆ’||๐บ2๎€ท๐บโˆฉโ„Ž1๎€ธโˆช๐พ๎€ท๎€ท2โงต๐บ2๎€ธ๎€ท๐บโˆฉโ„Ž1||=||โ„Ž๎€ท๐บ๎€ธ๎€ธ1๎€ธ||โˆ’||๐พ2๎€ท๐บโˆฉโ„Ž1๎€ธ||=||๐น2||.(2.4) Moreover, โ„Ž๎€ท๐พ1โˆช๐น1๎€ธ๎€ท๐พ=โ„Ž1๎€ธ๎€ท๐นโˆชโ„Ž1๎€ธ๎€ท๐พ=โ„Ž1๎€ธโˆช๎€ท๐บ2๎€ท๐พโงตโ„Ž1๎€ท๐พ๎€ธ๎€ธ=โ„Ž1๎€ธโˆช๐บ2๐พ=โ„Ž๎€ท๎€ท1โงต๐บ1๎€ธโˆช๐บ1๎€ธโˆช๐บ2=๐พ2๎€ท๐บโˆชโ„Ž1๎€ธ=๐พ2โˆช๎€ทโ„Ž๎€ท๐บ1๎€ธโงต๐พ2๎€ธ=๐พ2โˆช๐น2.(2.5) Hence, (๐‘‹,๐œ) is an SACDH space.
Conversely, assume that (๐‘‹,๐œ) is an SACDH space. Let ๐พ1,๐พ2 be two countable dense sets, therefore, there are two finite sets ๐บ1,๐บ2 with |๐บ1|=|๐บ2| and โ„Žโˆˆ๐ป(๐‘‹) such that โ„Ž(๐พ1โˆช๐บ1)=๐พ2โˆช๐บ2; moreover, ๐พ1โˆฉ๐บ1=๐บ2โˆฉ๐พ2=โ„Ž(๐บ1)โˆฉ๐บ2=โˆ…. Let ๐น1=โ„Žโˆ’1(๐บ2) and ๐น2=โ„Ž(๐บ1). Then |๐น1|=|๐น2|, moreover, ๐น1โІ๐พ1 and ๐น2โІ๐พ2. We need to prove that โ„Ž(๐พ1โงต๐น1)=๐พ2โงต๐น2. Now, โ„Ž(๐พ1โงต๐น1)=โ„Ž(๐พ1โงตโ„Žโˆ’1(๐บ2))=โ„Ž(๐พ1)โงต๐บ2. Claim that โ„Ž(๐พ1)โงต๐บ2=๐พ2โงต๐น2. If ๐‘ฅโˆˆโ„Ž(๐พ1)โงต๐บ2, then ๐‘ฅโˆˆ(๐พ2โˆช๐บ2)โงต๐บ2=๐พ2. As ๐พ1โˆฉ๐บ1=โˆ…, ๐‘ฅโˆ‰โ„Ž(๐บ1)=๐น2, so ๐‘ฅโˆˆ๐พ2โงต๐น2. To prove the other inclusion, suppose that ๐‘ฅโˆˆ๐พ2โงต๐น2=๐พ2โงตโ„Ž(๐บ1)=โ„Ž(โ„Žโˆ’1(๐พ2)โงต๐บ1). Therefore, ๐‘ฅ=โ„Ž(๐‘ก), for some ๐‘กโˆˆโ„Žโˆ’1(๐พ2)โงต๐บ1, and then ๐‘กโˆˆ(๐พ1โˆช๐บ1)โงต๐บ1 so ๐‘กโˆˆ๐พ1. Also ๐‘กโˆ‰โ„Žโˆ’1(๐บ2), hence ๐‘ฅโˆˆโ„Ž(๐พ1)โงต๐บ2.

Consequently, we have the following result.

Corollary 2.16. Every ACDH-1 space is SACDH.

3. T-Homogeneous Spaces

A transposition on ๐‘‹ is a permutation on ๐‘‹ which exchanges the places of two elements ๐‘ฅ,๐‘ฆ, 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 (๐‘‹,๐œ) is called Transposition-Homogeneous (TH) space if every transposition on ๐‘‹ is a homeomorphism.

Proposition 3.2. A space (๐‘‹,๐œ) is (TH) if and only if, for any two finite subsets ๐น1,๐น2 with the same cardinality and for every โ„Žโˆˆ๐ป(๐‘‹), there exists โ„Ž๎…žโˆˆ๐ป(๐‘‹) such that (1)โ„Ž๎…ž(๐น1)=๐น2,(2)โ„Ž๎…ž(๐‘ฅ)=โ„Ž(๐‘ฅ), for every ๐‘ฅโˆ‰๐น1โˆชโ„Žโˆ’1(๐น2),(3)โ„Ž๎…ž(โ„Žโˆ’1(๐น2))=โ„Ž(๐น1).

Proof. Let (๐‘‹,๐œ) be a TH space and ๐น a finite subset of ๐‘‹. Let ๐œŽ be a permutation which fixes ๐‘‹โงต๐น, clearly ๐œŽ is a composition of finite transpositions which is a homeomorphism. Let ๐น1,๐น2 be two finite subsets with the same cardinality and โ„Žโˆˆ๐ป(๐‘‹), clearly |๐น1โงตโ„Žโˆ’1(๐น2)|=|โ„Žโˆ’1(๐น2)โงต๐น1| and ๐น=(๐น1โงตโ„Žโˆ’1(๐น2))โŠ”(โ„Žโˆ’1(๐น2)โงต๐น1) are a disjoint union of finite sets which is finite. Let ๐‘“โˆถ๐นโ†’๐น be a bijection such that ๐‘“(๐น1โงตโ„Žโˆ’1(๐น2))=โ„Žโˆ’1(๐น2)โงต๐น1. For each ๐‘ฅโˆˆ๐น1โงตโ„Žโˆ’1(๐น2), let ๐œŽ๐‘ฅ be a transposition on ๐‘‹ which transposes ๐‘ฅ and ๐‘“(๐‘ฅ). Now let ๐œŽ be the finite composition of the transpositions ๐œŽ๐‘ฅ,๐‘ฅโˆˆ๐น1โงตโ„Žโˆ’1(๐น2). Then ๐œŽ๐‘ฅ(๐‘ฆ)=๐‘ฆ for all ๐‘ฆโˆˆ๐‘‹โงต๐น, ๐œŽ(๐น1)=โ„Žโˆ’1(๐น2), and ๐œŽ(โ„Žโˆ’1(๐น2))=๐น1. Now โ„Ž๎…ž=โ„Žโˆ˜๐œŽ is the required function since โ„Ž๎…ž(๐‘ฅ)=โ„Ž(๐‘ฅ), for every ๐‘ฅโˆˆ๐‘‹โงต๐น.
The converse is obvious by choosing โ„Ž to be the identity and ๐น1={๐‘ฅ}, ๐น2={๐‘ฆ}.

Example 3.3. One can show that the spaces (๐‘‹,๐œind), (๐‘‹,๐œdis), (๐‘‹,๐œcof), and (๐‘‹,๐œcoc) are all TH spaces. However, the spaces (โ„,๐œ๐‘™.๐‘Ÿ) and (โ„,๐œ๐‘ข) 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 ๐‘‹ be such that 1<|๐‘‹|<โ„ต0, with the indiscrete topology. The space (๐‘‹,๐œ) is a TH-space and it is also ACDH-1, but it is not a CDH space as it is not a ๐‘‡1 space.

Theorem 3.5. If (๐‘‹,๐œ) is SACDH, TH space, then (๐‘‹,๐œ) is a CDH space.

Proof. If ๐ด, ๐ต are two countable dense subsets of ๐‘‹, then there exist two finite subsets ๐น1,๐น2 of ๐ด,๐ต, respectively with |๐น1|=|๐น2| and there is โ„Žโˆˆ๐ป(๐‘‹) such that โ„Ž(๐ดโงต๐น1)=๐ตโงต๐น2. As ๐‘‹ is a TH space, there exists โ„Ž๎…žโˆˆ๐ป(๐‘‹) such that โ„Ž๎…ž(๐น1)=๐น2, โ„Ž๎…ž(๐‘ฅ)=โ„Ž(๐‘ฅ), for all ๐‘ฅโˆ‰๐น1โˆชโ„Žโˆ’1(๐น2) and โ„Ž๎…ž(โ„Žโˆ’1(๐น2))=โ„Ž(๐น1). To show that โ„Ž๎…ž(๐ด)=๐ต we show first that ๐ดโˆฉโ„Žโˆ’1(๐น2)โІ๐น1. Suppose that there is ๐‘ฅโˆˆ(๐ดโˆฉโ„Žโˆ’1(๐น2))โงต๐น1. Then โ„Ž(๐‘ฅ)โˆˆ๐ตโงต๐น2 which gives a contradiction. Hence โ„Ž๎…ž(๐ดโงต๐น1)=โ„Ž(๐ดโงต๐น1)=๐ตโงต๐น2, so โ„Ž๎…ž(๐ด)=โ„Ž๎…ž((๐ดโงต๐น1)โˆช๐น1)=(๐ตโงต๐น2)โˆช๐น2=๐ต.

Let (๐‘‹,๐œ) be an ACDH-1 space and ๐”‰={๐นโˆถ๐นisarelatedset} and define ๐ท={๐ดโІ๐‘‹โˆถ๐ดisacountabledensesubsetof๐‘‹}, and also we define the relation โˆผ on ๐ท by: ๐ดโˆผ๐ต if and only if there is ๐นโˆˆ๐”‰ such that |๐ดโˆฉ๐น|=|๐ตโˆฉ๐น|. Let ๐ท1={(๐ด,๐ต)โˆˆ๐ทร—๐ทโˆถ๐ดโˆผ๐ต}.

Now we have the following result.

Theorem 3.6. If (๐‘‹,๐œ) is an ACDH-1, TH space, and (๐ด,๐ต)โˆˆ๐ท1, then there is โ„Žโˆˆ๐ป(๐‘‹) such that โ„Ž(๐ด)=๐ต.

Proof. Suppose that (๐ด,๐ต)โˆˆ๐ท1, so there are ๐นโˆˆ๐”‰ and โ„Žโˆˆ๐ป(๐‘‹) such that โ„Ž(๐ดโงต๐น)=๐ตโงต๐น with |๐ดโˆฉ๐น|=|๐ตโˆฉ๐น|. If ๐ดโˆฉ๐น=โˆ…, then we are done. In general let ๐น1=๐นโˆฉ๐ด and ๐น2=๐นโˆฉ๐ต, therefore |๐น1|=|๐น2|. As ๐‘‹ is a TH space, there exists โ„Ž๎…žโˆˆ๐ป(๐‘‹) such that โ„Ž๎…ž(๐น1)=๐น2andโ„Ž๎…ž(๐‘ฅ)=โ„Ž(๐‘ฅ), for all ๐‘ฅโˆ‰๐น1โˆชโ„Žโˆ’1(๐น2) and โ„Ž๎…ž(โ„Žโˆ’1(๐น2))=โ„Ž(๐น1), also ๐ดโˆฉโ„Žโˆ’1(๐น2)โІ๐น. Now โ„Ž๎…ž(๐ดโงตF1)=โ„Ž๎…ž(๐ดโงต๐น)=โ„Ž(๐ดโงต๐น)=๐ตโงต๐น=๐ตโงต๐น2, so โ„Ž๎…ž(๐ด)=โ„Ž๎…ž((๐ดโงต๐น1)โˆช๐น1)=(๐ตโงต๐น2)โˆช๐น2=๐ต.

Consequently, we have the following Corollary.

Corollary 3.7. If (๐‘‹,๐œ) is an ACDH-1, TH space, and ๐ท1=๐ทร—๐ท, then (๐‘‹,๐œ) is a CDH space.

4. Almost CDH Spaces, and New Separation Axioms

We know that almost CDH space is not a ๐‘‡0 space. In this section we will give a characterization of almost ๐‘‡0 space, also we will give a new definition of almost ๐‘‡1 space.

Theorem 4.1. Let (๐‘‹,๐œ) be a topological space. Then (๐‘‹,๐œ) is ๐ด๐‘‡0 space if and only if there exists a finite subset ๐น of ๐‘‹, such that, for all ๐‘ฅโ‰ ๐‘ฆ and {๐‘ฅ,๐‘ฆ}โˆฉ๐น=โˆ…, there is an open set containing only one of ๐‘ฅ, ๐‘ฆ. We will refer to ๐น by a related finite set ๐น.

Proof. Assume that (๐‘‹,๐œ) is an ๐ด๐‘‡0 space. Let ๐น=๐ด. Then for all ๐‘ฅโ‰ ๐‘ฆ with {๐‘ฅ,๐‘ฆ}โˆฉ๐น=โˆ…, we have that |๐ด๐‘ฅ|=1 and |๐ด๐‘ฆ|=1, hence ๐‘ฆโˆ‰๐‘ฅ or ๐‘ฅโˆ‰๐‘ฆ. Conversely, suppose that there exists a finite set ๐น such that for all ๐‘ฅโ‰ ๐‘ฆ with {๐‘ฅ,๐‘ฆ}โˆฉ๐น=โˆ…, there is an open set containing only one of ๐‘ฅ, ๐‘ฆ. If |๐ด| is an infinite, then there exists a denumerable subset of ๐ด, say {๐‘ฅ1,๐‘ฅ2,โ€ฆ}, and |๐ด๐‘ฅ๐‘›|>1, for all ๐‘›โˆˆโ„•, so there exist ๐‘ฆ๐‘›โˆˆ๐ด๐‘ฅ๐‘› and ๐‘ฆ๐‘›โ‰ ๐‘ฅ๐‘› for all ๐‘›โˆˆโ„•. Therefore there is ๐‘›0โˆˆโ„• such that ๐‘ฅ๐‘›0โ‰ ๐‘ฆ๐‘›0 are both not in ๐น and ๐‘ฅ๐‘›0=๐‘ฆ๐‘›0, which gives a contradiction.

Definition 4.2. A space (๐‘‹,๐œ) is called an almost strong ๐‘‡1 (๐ด๐‘†๐‘‡1) space if there is a finite subset ๐น of ๐‘‹ such that, for all ๐‘ฅโ‰ ๐‘ฆ and {๐‘ฅ,๐‘ฆ}โˆฉ๐น=โˆ…, there are two open subsets ๐‘ข1,๐‘ข2 of ๐‘‹, such that ๐‘ฅโˆˆ๐‘ข1โงต๐‘ข2 and ๐‘ฆโˆˆ๐‘ข2โงต๐‘ข1. ๐น is called the related finite set.

One may easily prove the following proposition.

Proposition 4.3. Let (๐‘‹,๐œ) be a topological space. If (๐‘‹,๐œ) is an ๐ด๐‘†๐‘‡1 space, then for all ๐‘ฅโˆ‰๐น, we have ๐‘ฅโІ{๐‘ฅ}โˆช๐น; where ๐น is a related finite set. Conversely, if there is a finite set ๐น such that for all ๐‘ฅโˆ‰๐น, ๐‘ฅโІ{๐‘ฅ}โˆช๐น, then (๐‘‹,๐œ) is an ๐ด๐‘†๐‘‡1.

In the following results we show that the new separation axiom ๐ด๐‘†๐‘‡1 is stronger than the one defined by Fora et al. in [8].

Proposition 4.4. Every ๐ด๐‘†๐‘‡1 space is ๐ด๐‘‡1 space.

Proof. Suppose that (๐‘‹,๐œ) is an ๐ด๐‘†๐‘‡1 space, and let ๐น be a related finite set. By Theorem 4.1 it is an ๐ด๐‘‡0, therefore, |๐ด|<โ„ต0. By Proposition 4.3, for all ๐‘ฅโˆ‰๐น, we have ๐‘ฅโІ{๐‘ฅ}โˆช๐น, so that ๐ตโІ๐น. If |๐ถ|โ‰ฅโ„ต0, then there is ๐‘ฅ๐‘›โˆˆ๐ถ such that for all ๐‘šโ‰ ๐‘›, ๐‘ฅ๐‘›โ‰ ๐‘ฅ๐‘š and |๐ถ๐‘ฅ๐‘›|โ‰ฅโ„ต0, for all ๐‘›โˆˆโ„•. Let ๐‘›1 be such that ๐‘ฅ๐‘›1โˆ‰๐น. Therefore |๐ถ๐‘ฅ๐‘›1|โ‰ฅโ„ต0, then there is ๐‘กโˆˆ๐ถ๐‘ฅ๐‘›1โงต๐น, so ๐‘ฅ๐‘›1โˆˆ๐‘ก, 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 ๐‘‹=โ„•,โˆถ๐›ฝ={{1},{1,2},{3},{3,4},{5},{5,6},โ€ฆ} so ๐›ฝ is a base for some topology on ๐‘‹. Note that for ๐‘›โˆˆโ„•, we have ๎ƒฏ๐‘›={๐‘›,๐‘›+1};๐‘›isodd,{๐‘›};๐‘›iseven.(4.1)
Therefore, for all ๐‘ฅโˆˆโ„•, ๐ด๐‘ฅ={๐‘ฅ}, hence ๐ด=โˆ…, ๐ต=โˆ… as for all ๐‘›โˆˆโ„•, |๐‘›|โ‰ค2. Now for ๐‘›โˆˆโ„•๐ถ๐‘›=๎ƒฏ{๐‘›โˆ’1,๐‘›};๐‘›iseven,{๐‘›};๐‘›isodd.(4.2) Therefore, ๐ถ=โˆ…, and then (๐‘‹,๐œ(๐›ฝ)) is ๐ด๐‘‡1 space. Let ๐น be any finite subset of ๐‘‹. Let ๐‘š=supโˆถ๐น, as 2๐‘š+2โˆˆCl{2๐‘š+1} and 2๐‘š+2,2๐‘š+1 are both not in ๐น, (๐‘‹,๐œ(๐›ฝ)) is not an ๐ด๐‘†๐‘‡1 space.

One may easily prove the following proposition.

Proposition 4.6. Every ๐‘†๐ด๐‘‡1 space is ๐ด๐‘†๐‘‡1 space.

Fitzpatrick et al. proved in [6] that every CDH space is a ๐‘‡1 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 ๐ด๐‘†๐‘‡1 space by using the idea of almost closurely ordered sets.

Definition 4.7 (see [7]). Let (๐‘‹,๐œ) be a topological space. A countable subset ๐พ of ๐‘‹ is said to have the closurely ordered property if there exists a numeration of ๐พ, say ๐พ={๐‘ฅ1,๐‘ฅ2,โ€ฆ} such that for all ๐‘›โ‰ฅ2, ๐‘ฅ๐‘›โˆ‰Cl{๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›โˆ’1}. The numeration {๐‘ฅ1,๐‘ฅ2,โ€ฆ} is called closurely ordered countable set.

Definition 4.8 (see [7]). A countable collection ๐”„ of subsets of ๐‘‹ is said to have the closurely ordered countable property if ๐”„ can be written as ๐”„={๐ด1,๐ด2,โ€ฆ}, where ๐ด๐‘›โ‹ƒโˆฉCl{๐‘›โˆ’1๐‘–=1๐ด๐‘–}=โˆ…. The form {๐ด1,๐ด2,โ€ฆ} is called closurely ordered countable family.

Theorem 4.9 (see [7]). Let (๐‘‹,๐œ) be a topological space and let ๐พ be any countable dense subset of ๐‘‹. Then there exists a countable dense subset ๐พ1 of ๐พ, such that ๐พ1 is closurely ordered countable set.

Theorem 4.10 (see [7]). Let (๐‘‹,๐œ) be a topological space, then, (i)if โ„Žโˆถ๐‘‹โ†’๐‘Œ is an injective open function and ๐พ has the closurely ordered property in ๐‘‹, then โ„Ž(๐พ) has the closurely ordered property in ๐‘Œ,(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 ๐พ in (๐‘‹,๐œ) is said to have the almost closurely ordered property if there is a finite set ๐น in ๐‘‹ such that ๐พโงต๐น has the closurely ordered property. If ๐พโงต๐น={๐‘ฅ1,๐‘ฅ2,โ€ฆ} is a closurely ordered set, then ๐พ is called almost closurely ordered set.

Proposition 4.12. If (๐‘‹,๐œ) is an ๐ด๐‘†๐‘‡1 space and F is a related finite set, then each doubleton {๐‘ฅ,๐‘ฆ}โІ๐‘‹โงต๐น has the closurely ordered property. Conversely, in a topological space (๐‘‹,๐œ) if there exists a finite set ๐น all doubletons {๐‘ฅ,๐‘ฆ}โІ๐‘‹โงต๐น have the closurely ordered property, then (๐‘‹,๐œ), is an ๐ด๐‘‡0 space.

Proof. The first part is clear. To prove the converse, assume that there is such a finite set ๐น. Let ๐‘ฅ,๐‘ฆ be such that ๐‘ฅโ‰ ๐‘ฆ and {๐‘ฅ,๐‘ฆ}โІ๐‘‹โงต๐น. So ๐‘ฅโˆ‰๐‘ฆ or ๐‘ฆโˆ‰๐‘ฅ, by Theorem 4.1 (๐‘‹,๐œ) is an ๐ด๐‘‡0.

Theorem 4.13. Every ACDH-1 space is ๐ด๐‘‡0 space.

Proof. Assume that (๐‘‹,๐œ) is an ACDH-1 space and ๐น is a related finite set. We want to show that ๐น is the desired set. If ๐‘ฅโ‰ ๐‘ฆ with {๐‘ฅ,๐‘ฆ}โˆฉ๐น=โˆ… and ๐พ is a countable dense subset of ๐‘‹, by Theorem 4.9, we may assume that ๐พ has the closurely ordered property, also by Theorem 4.10, ๐พโงต๐น has the closurely ordered property. Now {๐‘ฅ,๐‘ฆ}โˆช๐พ is also a countable dense subset of ๐‘‹, therefore there is โ„Žโˆˆ๐ป(๐‘‹) such that โ„Ž(๐พโงต๐น)=(Kโˆช{๐‘ฅ,๐‘ฆ})โงต๐น. So {๐‘ฅ,๐‘ฆ} has the closurely ordered property, the result follows by Theorem 4.10.

Theorem 4.14 (see [7]). If (๐‘‹,๐œ) a topological space and K is a countable dense subset, then there exists a countable collection of subsets ๐ด1,๐ด2,โ€ฆ of ๐พ such that (i)โ‹ƒ๐ด=๐‘š๐‘›=1๐ด๐‘›โІ๐พโˆถ๐‘šโ‰คโ„ต0,(ii){๐ด1,๐ด2,โ€ฆ} is closurely ordered countable family,(iii)each ๐ด๐‘˜ has the closurely ordered property,(iv)each ๐ด๐‘˜ is either a singleton or an infinite set,(v)๐ด=๐พ,(vi)if ๐ด๐‘˜ is a singleton, say {๐‘Ž๐‘˜}, then ๐‘Ž๐‘˜โˆ‰๐‘ฅ and ๐‘ฅโˆ‰Cl{๐‘Ž๐‘˜}, for all ๐‘ฅโˆˆ๐ดโงต{๐‘Ž๐‘˜},(vii)if ๐ด๐‘˜={๐‘Ž๐‘˜1,๐‘Ž๐‘˜2,โ€ฆ} is infinite set, then ๐‘Ž๐‘˜๐‘–โˆˆCl{๐‘Ž๐‘˜1+๐‘–}, for all ๐‘–.

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 (๐‘‹,๐œ) be an ACDH-1 space and let ๐น be a related finite set. If ๐‘ฅโ‰ ๐‘ฆand{๐‘ฅ,๐‘ฆ}โˆฉ๐น=โˆ… and ๐‘ฅโˆˆ๐‘ฆ, then ๐‘ฅ is an infinite set.

Proof. Suppose that ๐‘ฅโ‰ ๐‘ฆ,{๐‘ฅ,๐‘ฆ}โˆฉ๐น=โˆ… and ๐‘ฅโˆˆ๐‘ฆ. Suppose that ๐‘ฅ is a finite set, then there is ๐‘โˆˆโ„• such that |๐‘ฅ|=๐‘. Let ๐พ be any countable dense set in ๐‘‹, then by Theorem 4.14, there is a countable collection of subsets of ๐พ say ๐ด1,๐ด2,โ€ฆ satisfing the conditions (i)โ€“(vii). So, โ‹ƒ๐ด=๐‘š๐‘›=1๐ด๐‘›โІ๐พ and ๐ด=๐พ=๐‘‹. Let ๐ผ={๐‘–โˆถ|๐ด๐‘–|=โ„ต๐‘œ}. For ๐‘–โˆˆ๐ผ, define ๐ต๐‘–=๐ด๐‘–โงต{๐‘Ž๐‘–1,๐‘Ž๐‘–2,โ€ฆ,๐‘Ž๐‘–๐‘} and for ๐‘–โˆ‰๐ผ๐ต๐‘–=๐ด๐‘–={๐‘Ž๐‘–}, also define โ‹ƒ๐ต=๐‘š๐‘–=1๐ต๐‘–. To show that ๐ต=๐ด. If ๐‘–โˆˆ๐ผ, then Cl{๐‘Ž๐‘˜1,๐‘Ž๐‘˜2,โ€ฆ,๐‘Ž๐‘˜๐‘}โІCl{๐‘Ž๐‘–๐‘+1}โІ๐ต๐‘–, therefore ๐ด๐‘–โІ๐ต๐‘–โˆชCl{๐‘Ž๐‘–1,๐‘Ž๐‘–2,โ€ฆ,๐‘Ž๐‘–๐‘}โІ๐ต๐‘–. Then ๐ต is a countable dense set in ๐‘‹ and so is ๐ตโˆช{๐‘ฅ,๐‘ฆ}. Therefore, there exists โ„Žโˆˆ๐ป(๐‘‹) such that โ„Ž((๐ตโˆช{๐‘ฅ,๐‘ฆ}โงต๐น))=๐ตโงต๐น. As ๐‘ฅโˆ‰๐น, we have โ„Ž(๐‘ฅ)โˆˆ๐ต๐‘– for some ๐‘–โˆˆโ„•. If |๐ต๐‘–|=1, then ๐ต๐‘–={๐‘๐‘–} and โ„Ž(๐‘ฅ)=๐‘๐‘–, so ๐‘๐‘–โˆˆโ„Ž(๐‘ฆ) where ๐‘ฆโˆ‰๐น; therefore, โ„Ž(๐‘ฆ)โˆˆ๐ตโงต{๐‘๐‘–} which is a contradiction by Theorem 4.14(vi). If |๐ต๐‘–|=โ„ต0 and โ„Ž(๐‘ฅ)=๐‘๐‘–๐‘›๐‘œ for some ๐‘›๐‘œ>๐‘, so all ๐‘๐‘˜๐‘–,๐‘๐‘–2,โ€ฆ,๐‘๐‘–๐‘› are in โ„Ž(๐‘ฅ). As |โ„Ž(๐‘ฅ)|=|๐‘ฅ|>๐‘, which is impossible, so ๐‘ฅ is an infinite set.

Recall that in ACDH-1 space, if ๐พ is a countable dense subset of ๐‘‹, then by Theorem 4.14 there are countable subsets ๐ด1,๐ด2,โ€ฆ of ๐พ satisfying (i)โ€“(vii) of the pointed theorem.

Moreover, โ‹ƒ๐ด=๐‘š๐‘–=1๐ด๐‘–โІ๐พ, ๐ด=๐พ=๐‘‹. Therefore, there is โ„Žโˆˆ๐ป(๐‘‹) such that โ„Ž(๐ดโงต๐น)=๐พโงต๐น. By Proposition 4.15, ๐พโงต๐น can be decomposed in the same way as ๐ดโงต๐น.

The following theorem shows that all the above ๐ด๐‘–s are singletons.

Theorem 4.17. Let (๐‘‹,๐œ) be an ACDH-1 space, K any countable dense subset of ๐‘‹, and ๐น a related finite set. Then โ‹ƒ๐พโงต๐น=โˆž๐‘–=1๐ด๐‘– and |๐ด๐‘–|=1.

Proof. Let ๐ผ={๐‘–โˆถ|๐ด๐‘–|=โ„ต0}. If ๐ผโ‰ โˆ…, then ๐ด๐‘–={๐‘Ž๐‘–1,๐‘Ž๐‘–2,โ€ฆ}, for some ๐‘–โˆˆ๐ผ. We have ๐‘Ž๐‘–1โˆˆCl{๐‘Ž๐‘–2} and {๐‘Ž๐‘–1,๐‘Ž๐‘–2}โˆฉ๐น=โˆ…, so, by Theorem 4.16, we have Cl{๐‘Ž๐‘–1} which is an infinite set. Let ๐‘Ž๐‘–0โˆ‰๐น with ๐‘Ž๐‘–0โˆˆCl{๐‘Ž๐‘–1}โงต{๐‘Ž๐‘–1}. In a similar way, let ๐‘Ž๐‘–โˆ’1โˆ‰๐น with ๐‘Ž๐‘–โˆ’1โˆˆCl{๐‘Ž๐‘–0}โงต{๐‘Ž๐‘–0}. By the same argument, we have a sequence โ€ฆ,๐‘Ž๐‘–โˆ’๐‘›๐‘Ž๐‘–โˆ’๐‘›+1,โ€ฆ,๐‘Ž๐‘–โˆ’1,๐‘Ž๐‘–0 and ๐‘Ž๐‘–โˆ’๐‘˜โˆˆCl{๐‘Ž๐‘–โˆ’๐‘˜+1}โงต{๐‘Ž๐‘–โˆ’๐‘˜+1}. Now we claim that for all ๐‘˜,๐‘›โˆˆโ„•,๐‘Ž๐‘–โˆ’๐‘˜โ‰ ๐‘Ž๐‘–๐‘›. If ๐‘Ž๐‘–โˆ’๐‘˜=๐‘Ž๐‘–๐‘› for some ๐‘›โ‰ฅ2 and ๐‘˜โ‰ฅ0, then ๐‘Ž๐‘–๐‘›โˆˆCl{๐‘Ž๐‘–1} which contradicts the fact that {๐‘Ž๐‘–1,๐‘Ž๐‘–2,โ€ฆ} is a closurely ordered set. Also ๐‘Ž๐‘–1โ‰ ๐‘Ž๐‘–โˆ’๐‘˜ for all ๐‘˜>0, since ๐‘Ž๐‘–1โˆ‰Cl{๐‘Ž๐‘–0} and ๐‘Ž๐‘–โˆ’๐‘˜โˆˆCl{๐‘Ž๐‘–0}, so we proved our claim. Let ๐พ1โ‹ƒ=[๐‘–โˆ‰๐ผ๐ด๐‘–โ‹ƒ]โˆช[๐‘–โˆˆ๐ผ{โ€ฆ,๐‘Ž๐‘–โˆ’2๐‘Ž๐‘–โˆ’1,๐‘Ž๐‘–0,๐‘Ž๐‘–1,๐‘Ž๐‘–1,โ€ฆ}]. Then ๐‘‹=๐พโІ๐พ1โˆช๐น, hence ๐พ1โˆช๐น is a countable dense subset of ๐‘‹. Then there is โ„Žโˆˆ๐ป(๐‘‹) such that โ„Ž(๐พโงต๐น)=(๐พ1โˆช๐น)โงต๐น=๐พ1. For ๐‘–โˆ‰๐ผ,๐ด๐‘–={๐‘Ž๐‘–} and ๐‘Ž๐‘–โˆ‰๐‘ฅ, for all ๐‘ฅโˆˆ(๐พโงต๐น)โงต{๐‘Ž๐‘–}. Then by Proposition 4.15, โ„Ž(๐ด๐‘–)={๐‘Ž๐‘—}=๐ด๐‘—, ๐‘—โˆ‰๐ผ. Now define ๐‘–0=inf(๐ผ), therefore ๐ด๐‘–0={๐‘Ž๐‘–01,๐‘Ž๐‘–02,โ€ฆ}. Moreover, โ„Ž(๐‘Ž๐‘–01)โˆˆ๐ด๐‘—0, for some ๐‘—๐‘œโˆˆ๐ผ, where ๐ด๐‘—0={โ€ฆ,๐‘Ž๐‘—0โˆ’2,๐‘Ž๐‘—0โˆ’1,โ‹ฏ,๐‘Ž๐‘—01,๐‘Ž๐‘—02,โ€ฆ}. Then โ„Ž(๐‘Ž๐‘–01)=๐‘Ž๐‘—0๐‘˜, for some ๐‘˜โˆˆโ„ค. Also ๐‘ฅโˆˆCl{๐‘Ž๐‘–01}, where ๐‘ฅโˆˆ๐พโงต๐น and ๐‘ฅ=โ„Žโˆ’1(๐‘Ž๐‘—0๐‘˜โˆ’1). If ๐‘ฅโˆˆ๐ด๐‘Ÿ, for some ๐‘Ÿโˆˆ๐ผ, then ๐ด๐‘Ÿโ‹ƒโˆฉCl{๐‘Ÿโˆ’1๐‘—=1๐ด๐‘—}โ‰ โˆ… which is a contradiction, so ๐ผ=๐œ™.

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

Corollary 4.18. If (๐‘‹,๐œ) is an ACDH-1 space, ๐น is a related finite set, and ๐พ is a countable dense subset of ๐‘‹, then ๐พโงต๐น has the closurely ordered property.

Proof. If ๐พ is a countable dense subset of ๐‘‹, then by using Theorem 4.17, we have that โ‹ƒ๐พโงต๐น=โˆž๐‘–=1๐ด๐‘– and |๐ด๐‘–|=1, for all ๐‘–โˆˆโ„•. Therefore,โ‹ƒ๐พโงต๐น=โˆž๐‘–=1๐‘Ž๐‘– indeed, ๐‘Ž๐‘–โˆ‰Cl{๐‘Ž๐‘—}, for all ๐‘–โ‰ ๐‘—.

Corollary 4.19. Every ACDH-1 space is an ๐ด๐‘†๐‘‡1 space.

Proof. Let ๐น be a related finite set and ๐‘ฅโ‰ ๐‘ฆ with {๐‘ฅ,๐‘ฆ}โˆฉ๐น=โˆ…. If ๐พ is a countable dense subset of ๐‘‹, say ๐พ={๐‘ฅ1,๐‘ฅ2,โ€ฆ}, then the set ๐พ1={๐‘ฅ,๐‘ฆ,๐‘ฅ1,๐‘ฅ2,โ€ฆ} is also a countable dense subset of ๐‘‹, therefore by Corollary 4.18, we have that ๐พ1โงต๐น has the closurely ordered property and {๐‘ฅ,๐‘ฆ}โІ๐พ1โงต๐น, therefore ๐‘ฆโˆ‰๐‘ฅ. Similarly, ๐‘ฅโˆ‰๐‘ฆ.

Acknowledgment

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