International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2010 / Article

Research Article | Open Access

Volume 2010 |Article ID 375293 | 11 pages | https://doi.org/10.1155/2010/375293

New Types of Almost Countable Dense Homogeneous Space

Academic Editor: Richard Wilson
Received28 Sep 2009
Revised19 Jan 2010
Accepted09 Mar 2010
Published05 May 2010

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 𝑇1 space. Afterward Bsoul, Fora, and Tallafha gave another proof for the same result, also they defined the almost CDH spaces and almost 𝑇1, 𝑇0 spaces, indeed they prove that every ACDH space is an almost 𝑇1 space. In this paper we introduce a new type of almost CDH spaces called ACDH-1, we characterize the ACDH spaces, the almost 𝑇0 spaces, we also give relations between different types of CDH spaces. We define new type of almost 𝑇1 (𝐴𝑇1) 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 𝐴𝑇1.

1. Introduction

In 1920, Sierpinski introduced in [1] the notion of homogeneous spaces, saying that a topological space 𝑋 is a homogeneous space if for any π‘₯≠𝑦 in 𝑋, then there is a homeomorphism β„Ž of 𝑋 such that β„Ž(π‘₯)=𝑦. FrΓ©chet in [2] and Brouwer in [3] observed that the 𝑛-dimensional Euclidean space ℝ𝑛 has the property that if 𝐴,𝐡 are countable dense subsets of ℝ𝑛, then there is a homeomorphism β„Ž of ℝ𝑛 such that β„Ž(𝐴)=𝐡. 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 𝑇1 spaces, and afterward Tallafha et al. in [7] gave another proof for the fact that CDH spaces are 𝑇1 spaces. In [8], Fora et al. defined almost CDH (ACDH) spaces, almost 𝑇0,𝑇1 (𝐴𝑇0,𝐴𝑇1) 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 𝑇0,𝑇1 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 𝑇0 we introduce new definition of almost 𝑇1 space 𝐴𝑆𝑇1, 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 𝐴𝑆𝑇1 space.

Finally, the following abbreviations and symbols will be used throughout this paper. For a subset 𝐴 of a topological space (𝑋,𝜏), we write 𝐴, or Cl(𝐴) for the closure of 𝐴, for π‘₯βˆˆπ‘‹, by π‘₯ which denotes the closure of {π‘₯} and |𝐴| which denotes the cardinality of 𝐴. By 𝐻(𝑋) we mean the set of all homeomorphisms of 𝑋, Ξ” refers to the symmetric difference of sets, (𝑋,𝜏ind), (𝑋,𝜏dis), (𝑋,𝜏cof), and (𝑋,𝜏coc) denote the set 𝑋 with the indiscrete, discrete, cofinite, and cocountable topologies, and (ℝ,πœπ‘™β‹…π‘Ÿ),(ℝ,πœπ‘Ÿβ‹…π‘Ÿ), and (ℝ,πœπ‘’) denote ℝ, 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 (𝑋,𝜏) is called an almost countable dense homogeneous (ACDH) space if it is a separable space and for any two countable dense subsets 𝐾1,𝐾2, there are two finite subsets 𝐹1,𝐹2, 𝐾1∩𝐹1=𝐾2∩𝐹2=βˆ… and β„Žβˆˆπ»(𝑋) such that β„Ž(𝐾1βˆͺ𝐹1)=𝐾2βˆͺ𝐹2 and β„Ž(𝐹1)∩𝐹2=βˆ…. In addition, if |𝐹1|=|𝐹2|, then (𝑋,𝜏) is called a strong ACDH space which is denoted by SACDH.

Remark 2.2. In the previous definition the condition β„Ž(𝐹1)∩𝐹2=βˆ… is redundant since if there exist finite sets 𝐡1 and 𝐡2 such that 𝐾1∩𝐡1=𝐾2∩𝐡2=βˆ… and β„Ž(𝐾1βˆͺ𝐡1)=𝐾2βˆͺ𝐡2, then we choose the finite sets 𝐹1 and 𝐹2 as follows 𝐹1=𝐡1β§΅β„Žβˆ’1(𝐢),𝐹2=𝐡2⧡𝐢, where 𝐢=β„Ž(𝐡1)∩𝐡2.

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

Let (𝑋,𝜏) be any topological space. For π‘₯βˆˆπ‘‹, let 𝐴π‘₯={π‘¦βˆˆπ‘‹βˆΆπ‘¦=π‘₯}, 𝐴={π‘₯∢|𝐴π‘₯|>1}, 𝐡={π‘₯∢|π‘₯|β‰₯β„΅0}, 𝐢π‘₯={π‘¦βˆˆπ‘‹βˆΆπ‘₯βˆˆπ‘¦}, and 𝐢={π‘₯∢|𝐢π‘₯|β‰₯β„΅0}. And Let 𝐹={π‘₯βˆˆπ‘‹βˆΆ{π‘₯}isnotaclosedset}.

Definition 2.4 (see [8]). A topological space (𝑋,𝜏) is called almost π‘‡π‘œ (𝐴𝑇0 ) if |𝐴|<β„΅0. If, in addition, |𝐡|<β„΅0, then (𝑋,𝜏) is called strong almost π‘‡π‘œ (π‘†π΄π‘‡π‘œ).

Clearly every π‘‡π‘œ-space is π΄π‘‡π‘œ.

Definition 2.5 (see [8]). A topological space (𝑋,𝜏) is called almost 𝑇1 and denoted by 𝐴𝑇1 if it is π‘†π΄π‘‡π‘œ and |𝐢|<β„΅0, that is, 𝐴,𝐡,and𝐢 are all finite sets.

Definition 2.6 (see [8]). A topological space (𝑋,𝜏) is called strong almost 𝑇1 space (𝑆𝐴𝑇1) if |𝐹|<β„΅π‘œ. Clearly every 𝑇1 space is 𝑆𝐴𝑇1 and every 𝑆𝐴𝑇1 is 𝐴𝑇1.

Theorem 2.7 (see [8]). Every ACDH space is 𝐴𝑇0.

Theorem 2.8 (see [8]). Let (𝑋,𝜏) be ACDH space. If |𝑋|≀℡0, then (𝑋,𝜏) is 𝐴𝑇1.

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

Definition 2.9. A topological space (𝑋,𝜏) is called an almost CDH of type 1 (ACDH-1) if it is separable space and there exists a finite subset 𝐹 such that, for any two countable dense subsets 𝐴 and 𝐡 of 𝑋, there exists β„Žβˆˆπ»(𝑋) such that β„Ž(𝐴⧡𝐹)=𝐡⧡𝐹. 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 (𝑋,𝜏) is ACDH-1, |𝑋|≀℡0, and F is a related finite set, then for all π‘₯βˆ‰πΉ, we have {π‘₯} which is an open set in X.

Proof. Let π‘₯βˆˆπ‘‹β§΅πΉ. If {π‘₯} is not an open set in 𝑋, then 𝑋⧡{π‘₯}=𝑋, therefore there is β„Žβˆˆπ»(𝑋) such that β„Ž(𝑋⧡𝐹)=(𝑋⧡{π‘₯})⧡𝐹 which is a contradiction.

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

Corollary 2.11. If (𝑋,𝜏) is an ACDH-1, |𝑋|≀℡0, then there exists the smallest related finite set 𝐴.

Proof. Let (𝑋,𝜏) be a nondiscrete ACDH-1 space and 𝐴={π‘₯βˆˆπ‘‹βˆΆ{π‘₯} not open}. As πœβ‰ πœdis, π΄β‰ πœ™. Also by Proposition 2.10, π΄βŠ†πΉ, for all related finite sets 𝐹, so 𝐴 is finite. If 𝐾1,𝐾2 are two dense sets and β„Žβˆˆπ»(𝑋), then π‘‹β§΅π΄βŠ†πΎ1∩𝐾2 and β„Ž(𝑋⧡𝐴)=𝑋⧡𝐴, hence β„Ž(𝐾1⧡𝐴)=β„Ž(𝑋⧡𝐴)=𝑋⧡𝐴=𝐾2⧡𝐴; therefore 𝐴 is the smallest finite related set. Moreover, if 𝜏=𝜏dis, then 𝐴=βˆ….

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

Example 2.12. Let 𝑋=𝐍 and 𝛽={{1,2},{3,4},{5},{6},…}. Let 𝐴,𝐡 be two dense subsets of 𝑋. Then {5,6,7,…}βŠ†π΄βˆ©π΅ let 𝐹={1,2,3,4}, hence (𝑋,𝜏(𝛽)) is ACDH-1 space which is not a CDH space as it is not a 𝑇1 space.

Now let prove the following characterizations of ACDH spaces.

Theorem 2.13. If (𝑋,𝜏) is a separable space, then the following are equivalent (i)(𝑋,𝜏) is an ACDH space(ii)For any two countable dense subsets 𝐾1,𝐾2, there exist two finite subsets 𝐹1,𝐹2 of 𝐾1,𝐾2, respectively, and β„Žβˆˆπ»(𝑋) such that β„Ž(𝐾1⧡𝐹1)=𝐾2⧡𝐹2.(iii)For any two countable dense subsets 𝐾1,𝐾2, there exist two equipotent finite subsets 𝐹1,𝐹2, and β„Žβˆˆπ»(𝑋) such that β„Ž(𝐾1⧡𝐹1)=𝐾2⧡𝐹2.(iv)For any two countable dense subsets 𝐾1,𝐾2, there exist two finite subsets 𝐹1,𝐹2, and β„Žβˆˆπ»(𝑋) such that β„Ž(𝐾1⧡𝐹1)=𝐾2⧡𝐹2.

Proof. (i) implies (ii) Suppose that (𝑋,𝜏) is ACDH space. Let 𝐾1,𝐾2 be two countable dense sets. Then there exist two finite sets 𝐺1,𝐺2 and β„Žβˆˆπ»(𝑋) such that 𝐺1∩𝐾1=𝐺2∩𝐾2=𝐺2βˆ©β„Ž(𝐺1)=βˆ… and β„Ž(𝐾1βˆͺ𝐺1)=𝐾2βˆͺ𝐺2. Let 𝐹1=β„Žβˆ’1(𝐺2)and𝐹2=β„Ž(𝐺1), so 𝐹1,𝐹2 are two finite sets and β„Žξ€·πΎ1⧡𝐹1ξ€Έ=ξ€·β„Žξ€·πΎ1βˆͺ𝐺1ξ€Έξ€·πΊβ§΅β„Ž1⧡𝐺2=𝐾2⧡𝐹2.(2.1) (ii) implies (iii) Let 𝐾1,𝐾2 be two countable dense sets, then there exist two finite sets 𝐹1,𝐹2, and β„Žβˆˆπ»(𝑋) such that 𝐹1βŠ†πΎ1,𝐹2βŠ†πΎ2, and β„Ž(𝐾1⧡𝐹1)=𝐾2⧡𝐹2. Let 𝐺1=𝐹1βˆͺβ„Žβˆ’1(𝐹2) and 𝐺2=β„Ž(𝐹1)βˆͺ𝐹2. Clearly |𝐺1|=|𝐺2|. Moreover, β„Žξ€·πΎ1⧡𝐺1𝐾=β„Ž1⧡𝐹1βˆͺβ„Žβˆ’1𝐹2ξ€Έξ€Έ=𝐾2⧡𝐹2.(2.2) But clearly (β„Ž(𝐹1)⧡𝐹2)∩𝐾2=βˆ…, so 𝐾2⧡𝐹2=𝐾2⧡𝐺2. (iii) implies that (iv) is clear.
(iv) implies (i) Let 𝐾1,𝐾2 be two countable dense sets, then there are 𝐺1,𝐺2 finite sets and β„Žβˆˆπ»(𝑋) such that β„Ž(𝐾1⧡𝐺1)=𝐾2⧡𝐺2.
Let 𝐹1=β„Žβˆ’1(𝐺2∩𝐾2)⧡(𝐺1∩𝐾1) and 𝐹2=β„Ž(𝐺1∩𝐾1)⧡(𝐺2∩𝐾2). Then 𝐹1,𝐹2 are two finite sets. If π‘‘βˆˆπΉ1∩𝐾1, then π‘‘βˆˆπΎ1⧡𝐺1 and β„Ž(𝑑)∈𝐺2∩𝐾2 which gives a contradiction. Similarly, 𝐹2∩𝐾2=βˆ…. Moreover, β„Žξ€·πΎ1βˆͺ𝐹1𝐾=β„Ž1ξ€Έβˆͺ𝐺2∩𝐾2ξ€Έξ€·πΊβ§΅β„Ž1∩𝐾1𝐾=β„Ž1ξ€Έβˆͺ𝐺2∩𝐾2𝐾=β„Žξ€·ξ€·1⧡𝐺1ξ€Έβˆͺ𝐾1∩𝐺1βˆͺ𝐺2∩𝐾2ξ€Έ=𝐾2⧡𝐺2𝐾βˆͺβ„Ž1∩𝐺1ξ€Έβˆͺ𝐺2∩𝐾2ξ€Έ=𝐾2βˆͺξ€·β„Žξ€·πΎ1∩𝐺1⧡𝐾2ξ€Έ=𝐾2βˆͺ𝐹2.(2.3)

Consequently, we have the following result.

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

Theorem 2.15. Let (𝑋,𝜏) be a separable space. Then (𝑋,𝜏) is an SACDH space if and only if for every two countable dense sets 𝐾1,𝐾2, there exist two equipotent finite subsets 𝐹1,𝐹2 of 𝐾1,𝐾2, respectively, and β„Žβˆˆπ»(𝑋) such that β„Ž(𝐾1⧡𝐹1)=𝐾2⧡𝐹2.

Proof. Let 𝐾1,𝐾2 be two countable dense sets, so there exist β„Žβˆˆπ»(𝑋), 𝐺1βŠ†πΎ1, 𝐺2βŠ†πΎ2, and |𝐺1|=|𝐺2|<β„΅0 such that β„Ž(𝐾1⧡𝐺1)=𝐾2⧡𝐺2. Let 𝐹1=β„Žβˆ’1(𝐺2)⧡𝐾1 and 𝐹2=β„Ž(𝐺1)⧡𝐾2. It is clear that 𝐹1∩𝐾1=𝐹2∩𝐾2=βˆ…. Also, ||𝐹1||=||β„Žβˆ’1𝐺2⧡𝐾1||=||β„Žβˆ’1𝐺2⧡𝐾1βˆ©β„Žβˆ’1𝐺2||=||β„Žξ€Έξ€Έβˆ’1𝐺2ξ€Έ||βˆ’||𝐾1βˆ©β„Žβˆ’1𝐺2ξ€Έ||=||𝐺1||βˆ’||ξ€·β„Žξ€·πΎ1⧡𝐺1𝐺βˆͺβ„Ž1ξ€Έξ€Έβˆ©πΊ2||=||𝐺1||βˆ’||β„Žξ€·πΊ1ξ€Έβˆ©πΊ2||=||𝐺1||βˆ’||𝐺2ξ€·πΊβˆ©β„Ž1ξ€Έβˆͺ𝐾2⧡𝐺2ξ€Έξ€·πΊβˆ©β„Ž1||=||β„Žξ€·πΊξ€Έξ€Έ1ξ€Έ||βˆ’||𝐾2ξ€·πΊβˆ©β„Ž1ξ€Έ||=||𝐹2||.(2.4) Moreover, β„Žξ€·πΎ1βˆͺ𝐹1𝐾=β„Ž1𝐹βˆͺβ„Ž1𝐾=β„Ž1ξ€Έβˆͺ𝐺2ξ€·πΎβ§΅β„Ž1𝐾=β„Ž1ξ€Έβˆͺ𝐺2𝐾=β„Žξ€·ξ€·1⧡𝐺1ξ€Έβˆͺ𝐺1ξ€Έβˆͺ𝐺2=𝐾2𝐺βˆͺβ„Ž1ξ€Έ=𝐾2βˆͺξ€·β„Žξ€·πΊ1⧡𝐾2ξ€Έ=𝐾2βˆͺ𝐹2.(2.5) Hence, (𝑋,𝜏) is an SACDH space.
Conversely, assume that (𝑋,𝜏) is an SACDH space. Let 𝐾1,𝐾2 be two countable dense sets, therefore, there are two finite sets 𝐺1,𝐺2 with |𝐺1|=|𝐺2| and β„Žβˆˆπ»(𝑋) such that β„Ž(𝐾1βˆͺ𝐺1)=𝐾2βˆͺ𝐺2; moreover, 𝐾1∩𝐺1=𝐺2∩𝐾2=β„Ž(𝐺1)∩𝐺2=βˆ…. Let 𝐹1=β„Žβˆ’1(𝐺2) and 𝐹2=β„Ž(𝐺1). Then |𝐹1|=|𝐹2|, moreover, 𝐹1βŠ†πΎ1 and 𝐹2βŠ†πΎ2. We need to prove that β„Ž(𝐾1⧡𝐹1)=𝐾2⧡𝐹2. Now, β„Ž(𝐾1⧡𝐹1)=β„Ž(𝐾1β§΅β„Žβˆ’1(𝐺2))=β„Ž(𝐾1)⧡𝐺2. Claim that β„Ž(𝐾1)⧡𝐺2=𝐾2⧡𝐹2. If π‘₯βˆˆβ„Ž(𝐾1)⧡𝐺2, then π‘₯∈(𝐾2βˆͺ𝐺2)⧡𝐺2=𝐾2. As 𝐾1∩𝐺1=βˆ…, π‘₯βˆ‰β„Ž(𝐺1)=𝐹2, so π‘₯∈𝐾2⧡𝐹2. To prove the other inclusion, suppose that π‘₯∈𝐾2⧡𝐹2=𝐾2β§΅β„Ž(𝐺1)=β„Ž(β„Žβˆ’1(𝐾2)⧡𝐺1). Therefore, π‘₯=β„Ž(𝑑), for some π‘‘βˆˆβ„Žβˆ’1(𝐾2)⧡𝐺1, and then π‘‘βˆˆ(𝐾1βˆͺ𝐺1)⧡𝐺1 so π‘‘βˆˆπΎ1. Also π‘‘βˆ‰β„Žβˆ’1(𝐺2), hence π‘₯βˆˆβ„Ž(𝐾1)⧡𝐺2.

Consequently, we have the following result.

Corollary 2.16. Every ACDH-1 space is SACDH.

3. T-Homogeneous Spaces

A transposition on 𝑋 is a permutation on 𝑋 which exchanges the places of two elements π‘₯,𝑦, 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 (𝑋,𝜏) is called Transposition-Homogeneous (TH) space if every transposition on 𝑋 is a homeomorphism.

Proposition 3.2. A space (𝑋,𝜏) is (TH) if and only if, for any two finite subsets 𝐹1,𝐹2 with the same cardinality and for every β„Žβˆˆπ»(𝑋), there exists β„Žξ…žβˆˆπ»(𝑋) such that (1)β„Žξ…ž(𝐹1)=𝐹2,(2)β„Žξ…ž(π‘₯)=β„Ž(π‘₯), for every π‘₯βˆ‰πΉ1βˆͺβ„Žβˆ’1(𝐹2),(3)β„Žξ…ž(β„Žβˆ’1(𝐹2))=β„Ž(𝐹1).

Proof. Let (𝑋,𝜏) be a TH space and 𝐹 a finite subset of 𝑋. Let 𝜎 be a permutation which fixes 𝑋⧡𝐹, clearly 𝜎 is a composition of finite transpositions which is a homeomorphism. Let 𝐹1,𝐹2 be two finite subsets with the same cardinality and β„Žβˆˆπ»(𝑋), clearly |𝐹1β§΅β„Žβˆ’1(𝐹2)|=|β„Žβˆ’1(𝐹2)⧡𝐹1| and 𝐹=(𝐹1β§΅β„Žβˆ’1(𝐹2))βŠ”(β„Žβˆ’1(𝐹2)⧡𝐹1) are a disjoint union of finite sets which is finite. Let π‘“βˆΆπΉβ†’πΉ be a bijection such that 𝑓(𝐹1β§΅β„Žβˆ’1(𝐹2))=β„Žβˆ’1(𝐹2)⧡𝐹1. For each π‘₯∈𝐹1β§΅β„Žβˆ’1(𝐹2), let 𝜎π‘₯ be a transposition on 𝑋 which transposes π‘₯ and 𝑓(π‘₯). Now let 𝜎 be the finite composition of the transpositions 𝜎π‘₯,π‘₯∈𝐹1β§΅β„Žβˆ’1(𝐹2). Then 𝜎π‘₯(𝑦)=𝑦 for all π‘¦βˆˆπ‘‹β§΅πΉ, 𝜎(𝐹1)=β„Žβˆ’1(𝐹2), and 𝜎(β„Žβˆ’1(𝐹2))=𝐹1. Now β„Žξ…ž=β„Žβˆ˜πœŽ is the required function since β„Žξ…ž(π‘₯)=β„Ž(π‘₯), for every π‘₯βˆˆπ‘‹β§΅πΉ.
The converse is obvious by choosing β„Ž to be the identity and 𝐹1={π‘₯}, 𝐹2={𝑦}.

Example 3.3. One can show that the spaces (𝑋,𝜏ind), (𝑋,𝜏dis), (𝑋,𝜏cof), and (𝑋,𝜏coc) are all TH spaces. However, the spaces (ℝ,πœπ‘™.π‘Ÿ) and (ℝ,πœπ‘’) 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 𝑋 be such that 1<|𝑋|<β„΅0, with the indiscrete topology. The space (𝑋,𝜏) is a TH-space and it is also ACDH-1, but it is not a CDH space as it is not a 𝑇1 space.

Theorem 3.5. If (𝑋,𝜏) is SACDH, TH space, then (𝑋,𝜏) is a CDH space.

Proof. If 𝐴, 𝐡 are two countable dense subsets of 𝑋, then there exist two finite subsets 𝐹1,𝐹2 of 𝐴,𝐡, respectively with |𝐹1|=|𝐹2| and there is β„Žβˆˆπ»(𝑋) such that β„Ž(𝐴⧡𝐹1)=𝐡⧡𝐹2. As 𝑋 is a TH space, there exists β„Žξ…žβˆˆπ»(𝑋) such that β„Žξ…ž(𝐹1)=𝐹2, β„Žξ…ž(π‘₯)=β„Ž(π‘₯), for all π‘₯βˆ‰πΉ1βˆͺβ„Žβˆ’1(𝐹2) and β„Žξ…ž(β„Žβˆ’1(𝐹2))=β„Ž(𝐹1). To show that β„Žξ…ž(𝐴)=𝐡 we show first that π΄βˆ©β„Žβˆ’1(𝐹2)βŠ†πΉ1. Suppose that there is π‘₯∈(π΄βˆ©β„Žβˆ’1(𝐹2))⧡𝐹1. Then β„Ž(π‘₯)∈𝐡⧡𝐹2 which gives a contradiction. Hence β„Žξ…ž(𝐴⧡𝐹1)=β„Ž(𝐴⧡𝐹1)=𝐡⧡𝐹2, so β„Žξ…ž(𝐴)=β„Žξ…ž((𝐴⧡𝐹1)βˆͺ𝐹1)=(𝐡⧡𝐹2)βˆͺ𝐹2=𝐡.

Let (𝑋,𝜏) be an ACDH-1 space and 𝔉={𝐹∢𝐹isarelatedset} and define 𝐷={π΄βŠ†π‘‹βˆΆπ΄isacountabledensesubsetof𝑋}, and also we define the relation ∼ on 𝐷 by: 𝐴∼𝐡 if and only if there is πΉβˆˆπ”‰ such that |𝐴∩𝐹|=|𝐡∩𝐹|. Let 𝐷1={(𝐴,𝐡)βˆˆπ·Γ—π·βˆΆπ΄βˆΌπ΅}.

Now we have the following result.

Theorem 3.6. If (𝑋,𝜏) is an ACDH-1, TH space, and (𝐴,𝐡)∈𝐷1, then there is β„Žβˆˆπ»(𝑋) such that β„Ž(𝐴)=𝐡.

Proof. Suppose that (𝐴,𝐡)∈𝐷1, so there are πΉβˆˆπ”‰ and β„Žβˆˆπ»(𝑋) such that β„Ž(𝐴⧡𝐹)=𝐡⧡𝐹 with |𝐴∩𝐹|=|𝐡∩𝐹|. If 𝐴∩𝐹=βˆ…, then we are done. In general let 𝐹1=𝐹∩𝐴 and 𝐹2=𝐹∩𝐡, therefore |𝐹1|=|𝐹2|. As 𝑋 is a TH space, there exists β„Žξ…žβˆˆπ»(𝑋) such that β„Žξ…ž(𝐹1)=𝐹2andβ„Žξ…ž(π‘₯)=β„Ž(π‘₯), for all π‘₯βˆ‰πΉ1βˆͺβ„Žβˆ’1(𝐹2) and β„Žξ…ž(β„Žβˆ’1(𝐹2))=β„Ž(𝐹1), also π΄βˆ©β„Žβˆ’1(𝐹2)βŠ†πΉ. Now β„Žξ…ž(𝐴⧡F1)=β„Žξ…ž(𝐴⧡𝐹)=β„Ž(𝐴⧡𝐹)=𝐡⧡𝐹=𝐡⧡𝐹2, so β„Žξ…ž(𝐴)=β„Žξ…ž((𝐴⧡𝐹1)βˆͺ𝐹1)=(𝐡⧡𝐹2)βˆͺ𝐹2=𝐡.

Consequently, we have the following Corollary.

Corollary 3.7. If (𝑋,𝜏) is an ACDH-1, TH space, and 𝐷1=𝐷×𝐷, then (𝑋,𝜏) is a CDH space.

4. Almost CDH Spaces, and New Separation Axioms

We know that almost CDH space is not a 𝑇0 space. In this section we will give a characterization of almost 𝑇0 space, also we will give a new definition of almost 𝑇1 space.

Theorem 4.1. Let (𝑋,𝜏) be a topological space. Then (𝑋,𝜏) is 𝐴𝑇0 space if and only if there exists a finite subset 𝐹 of 𝑋, such that, for all π‘₯≠𝑦 and {π‘₯,𝑦}∩𝐹=βˆ…, there is an open set containing only one of π‘₯, 𝑦. We will refer to 𝐹 by a related finite set 𝐹.

Proof. Assume that (𝑋,𝜏) is an 𝐴𝑇0 space. Let 𝐹=𝐴. Then for all π‘₯≠𝑦 with {π‘₯,𝑦}∩𝐹=βˆ…, we have that |𝐴π‘₯|=1 and |𝐴𝑦|=1, hence π‘¦βˆ‰π‘₯ or π‘₯βˆ‰π‘¦. Conversely, suppose that there exists a finite set 𝐹 such that for all π‘₯≠𝑦 with {π‘₯,𝑦}∩𝐹=βˆ…, there is an open set containing only one of π‘₯, 𝑦. If |𝐴| is an infinite, then there exists a denumerable subset of 𝐴, say {π‘₯1,π‘₯2,…}, and |𝐴π‘₯𝑛|>1, for all π‘›βˆˆβ„•, so there exist π‘¦π‘›βˆˆπ΄π‘₯𝑛 and 𝑦𝑛≠π‘₯𝑛 for all π‘›βˆˆβ„•. Therefore there is 𝑛0βˆˆβ„• such that π‘₯𝑛0≠𝑦𝑛0 are both not in 𝐹 and π‘₯𝑛0=𝑦𝑛0, which gives a contradiction.

Definition 4.2. A space (𝑋,𝜏) is called an almost strong 𝑇1 (𝐴𝑆𝑇1) space if there is a finite subset 𝐹 of 𝑋 such that, for all π‘₯≠𝑦 and {π‘₯,𝑦}∩𝐹=βˆ…, there are two open subsets 𝑒1,𝑒2 of 𝑋, such that π‘₯βˆˆπ‘’1⧡𝑒2 and π‘¦βˆˆπ‘’2⧡𝑒1. 𝐹 is called the related finite set.

One may easily prove the following proposition.

Proposition 4.3. Let (𝑋,𝜏) be a topological space. If (𝑋,𝜏) is an 𝐴𝑆𝑇1 space, then for all π‘₯βˆ‰πΉ, we have π‘₯βŠ†{π‘₯}βˆͺ𝐹; where 𝐹 is a related finite set. Conversely, if there is a finite set 𝐹 such that for all π‘₯βˆ‰πΉ, π‘₯βŠ†{π‘₯}βˆͺ𝐹, then (𝑋,𝜏) is an 𝐴𝑆𝑇1.

In the following results we show that the new separation axiom 𝐴𝑆𝑇1 is stronger than the one defined by Fora et al. in [8].

Proposition 4.4. Every 𝐴𝑆𝑇1 space is 𝐴𝑇1 space.

Proof. Suppose that (𝑋,𝜏) is an 𝐴𝑆𝑇1 space, and let 𝐹 be a related finite set. By Theorem 4.1 it is an 𝐴𝑇0, therefore, |𝐴|<β„΅0. By Proposition 4.3, for all π‘₯βˆ‰πΉ, we have π‘₯βŠ†{π‘₯}βˆͺ𝐹, so that π΅βŠ†πΉ. If |𝐢|β‰₯β„΅0, then there is π‘₯π‘›βˆˆπΆ such that for all π‘šβ‰ π‘›, π‘₯𝑛≠π‘₯π‘š and |𝐢π‘₯𝑛|β‰₯β„΅0, for all π‘›βˆˆβ„•. Let 𝑛1 be such that π‘₯𝑛1βˆ‰πΉ. Therefore |𝐢π‘₯𝑛1|β‰₯β„΅0, then there is π‘‘βˆˆπΆπ‘₯𝑛1⧡𝐹, so π‘₯𝑛1βˆˆπ‘‘, 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 𝑋=β„•,βˆΆπ›½={{1},{1,2},{3},{3,4},{5},{5,6},…} so 𝛽 is a base for some topology on 𝑋. Note that for π‘›βˆˆβ„•, we have 𝑛={𝑛,𝑛+1};𝑛isodd,{𝑛};𝑛iseven.(4.1)
Therefore, for all π‘₯βˆˆβ„•, 𝐴π‘₯={π‘₯}, hence 𝐴=βˆ…, 𝐡=βˆ… as for all π‘›βˆˆβ„•, |𝑛|≀2. Now for π‘›βˆˆβ„•πΆπ‘›=ξƒ―{π‘›βˆ’1,𝑛};𝑛iseven,{𝑛};𝑛isodd.(4.2) Therefore, 𝐢=βˆ…, and then (𝑋,𝜏(𝛽)) is 𝐴𝑇1 space. Let 𝐹 be any finite subset of 𝑋. Let π‘š=sup∢𝐹, as 2π‘š+2∈Cl{2π‘š+1} and 2π‘š+2,2π‘š+1 are both not in 𝐹, (𝑋,𝜏(𝛽)) is not an 𝐴𝑆𝑇1 space.

One may easily prove the following proposition.

Proposition 4.6. Every 𝑆𝐴𝑇1 space is 𝐴𝑆𝑇1 space.

Fitzpatrick et al. proved in [6] that every CDH space is a 𝑇1 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 𝐴𝑆𝑇1 space by using the idea of almost closurely ordered sets.

Definition 4.7 (see [7]). Let (𝑋,𝜏) be a topological space. A countable subset 𝐾 of 𝑋 is said to have the closurely ordered property if there exists a numeration of 𝐾, say 𝐾={π‘₯1,π‘₯2,…} such that for all 𝑛β‰₯2, π‘₯π‘›βˆ‰Cl{π‘₯1,π‘₯2,…,π‘₯π‘›βˆ’1}. The numeration {π‘₯1,π‘₯2,…} is called closurely ordered countable set.

Definition 4.8 (see [7]). A countable collection 𝔄 of subsets of 𝑋 is said to have the closurely ordered countable property if 𝔄 can be written as 𝔄={𝐴1,𝐴2,…}, where π΄π‘›β‹ƒβˆ©Cl{π‘›βˆ’1𝑖=1𝐴𝑖}=βˆ…. The form {𝐴1,𝐴2,…} is called closurely ordered countable family.

Theorem 4.9 (see [7]). Let (𝑋,𝜏) be a topological space and let 𝐾 be any countable dense subset of 𝑋. Then there exists a countable dense subset 𝐾1 of 𝐾, such that 𝐾1 is closurely ordered countable set.

Theorem 4.10 (see [7]). Let (𝑋,𝜏) be a topological space, then, (i)if β„ŽβˆΆπ‘‹β†’π‘Œ is an injective open function and 𝐾 has the closurely ordered property in 𝑋, then β„Ž(𝐾) has the closurely ordered property in π‘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 𝐾 in (𝑋,𝜏) is said to have the almost closurely ordered property if there is a finite set 𝐹 in 𝑋 such that 𝐾⧡𝐹 has the closurely ordered property. If 𝐾⧡𝐹={π‘₯1,π‘₯2,…} is a closurely ordered set, then 𝐾 is called almost closurely ordered set.

Proposition 4.12. If (𝑋,𝜏) is an 𝐴𝑆𝑇1 space and F is a related finite set, then each doubleton {π‘₯,𝑦}βŠ†π‘‹β§΅πΉ has the closurely ordered property. Conversely, in a topological space (𝑋,𝜏) if there exists a finite set 𝐹 all doubletons {π‘₯,𝑦}βŠ†π‘‹β§΅πΉ have the closurely ordered property, then (𝑋,𝜏), is an 𝐴𝑇0 space.

Proof. The first part is clear. To prove the converse, assume that there is such a finite set 𝐹. Let π‘₯,𝑦 be such that π‘₯≠𝑦 and {π‘₯,𝑦}βŠ†π‘‹β§΅πΉ. So π‘₯βˆ‰π‘¦ or π‘¦βˆ‰π‘₯, by Theorem 4.1 (𝑋,𝜏) is an 𝐴𝑇0.

Theorem 4.13. Every ACDH-1 space is 𝐴𝑇0 space.

Proof. Assume that (𝑋,𝜏) is an ACDH-1 space and 𝐹 is a related finite set. We want to show that 𝐹 is the desired set. If π‘₯≠𝑦 with {π‘₯,𝑦}∩𝐹=βˆ… and 𝐾 is a countable dense subset of 𝑋, by Theorem 4.9, we may assume that 𝐾 has the closurely ordered property, also by Theorem 4.10, 𝐾⧡𝐹 has the closurely ordered property. Now {π‘₯,𝑦}βˆͺ𝐾 is also a countable dense subset of 𝑋, therefore there is β„Žβˆˆπ»(𝑋) such that β„Ž(𝐾⧡𝐹)=(Kβˆͺ{π‘₯,𝑦})⧡𝐹. So {π‘₯,𝑦} has the closurely ordered property, the result follows by Theorem 4.10.

Theorem 4.14 (see [7]). If (𝑋,𝜏) a topological space and K is a countable dense subset, then there exists a countable collection of subsets 𝐴1,𝐴2,… of 𝐾 such that (i)⋃𝐴=π‘šπ‘›=1π΄π‘›βŠ†πΎβˆΆπ‘šβ‰€β„΅0,(ii){𝐴1,𝐴2,…} is closurely ordered countable family,(iii)each π΄π‘˜ has the closurely ordered property,(iv)each π΄π‘˜ is either a singleton or an infinite set,(v)𝐴=𝐾,(vi)if π΄π‘˜ is a singleton, say {π‘Žπ‘˜}, then π‘Žπ‘˜βˆ‰π‘₯ and π‘₯βˆ‰Cl{π‘Žπ‘˜}, for all π‘₯∈𝐴⧡{π‘Žπ‘˜},(vii)if π΄π‘˜={π‘Žπ‘˜1,π‘Žπ‘˜2,…} is infinite set, then π‘Žπ‘˜π‘–βˆˆCl{π‘Žπ‘˜1+𝑖}, for all 𝑖.

It is easy to prove the following result.

Proposition 4.15. The properties (i)–(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 (𝑋,𝜏) be an ACDH-1 space and let 𝐹 be a related finite set. If π‘₯≠𝑦and{π‘₯,𝑦}∩𝐹=βˆ… and π‘₯βˆˆπ‘¦, then π‘₯ is an infinite set.

Proof. Suppose that π‘₯≠𝑦,{π‘₯,𝑦}∩𝐹=βˆ… and π‘₯βˆˆπ‘¦. Suppose that π‘₯ is a finite set, then there is π‘βˆˆβ„• such that |π‘₯|=𝑁. Let 𝐾 be any countable dense set in 𝑋, then by Theorem 4.14, there is a countable collection of subsets of 𝐾 say 𝐴1,𝐴2,… satisfing the conditions (i)–(vii). So, ⋃𝐴=π‘šπ‘›=1π΄π‘›βŠ†πΎ and 𝐴=𝐾=𝑋. Let 𝐼={π‘–βˆΆ|𝐴𝑖|=β„΅π‘œ}. For π‘–βˆˆπΌ, define 𝐡𝑖=𝐴𝑖⧡{π‘Žπ‘–1,π‘Žπ‘–2,…,π‘Žπ‘–π‘} and for π‘–βˆ‰πΌπ΅π‘–=𝐴𝑖={π‘Žπ‘–}, also define ⋃𝐡=π‘šπ‘–=1𝐡𝑖. To show that 𝐡=𝐴. If π‘–βˆˆπΌ, then Cl{π‘Žπ‘˜1,π‘Žπ‘˜2,…,π‘Žπ‘˜π‘}βŠ†Cl{π‘Žπ‘–π‘+1}βŠ†π΅π‘–, therefore π΄π‘–βŠ†π΅π‘–βˆͺCl{π‘Žπ‘–1,π‘Žπ‘–2,…,π‘Žπ‘–π‘}βŠ†π΅π‘–. Then 𝐡 is a countable dense set in 𝑋 and so is 𝐡βˆͺ{π‘₯,𝑦}. Therefore, there exists β„Žβˆˆπ»(𝑋) such that β„Ž((𝐡βˆͺ{π‘₯,𝑦}⧡𝐹))=𝐡⧡𝐹. As π‘₯βˆ‰πΉ, we have β„Ž(π‘₯)βˆˆπ΅π‘– for some π‘–βˆˆβ„•. If |𝐡𝑖|=1, then 𝐡𝑖={𝑏𝑖} and β„Ž(π‘₯)=𝑏𝑖, so π‘π‘–βˆˆβ„Ž(𝑦) where π‘¦βˆ‰πΉ; therefore, β„Ž(𝑦)∈𝐡⧡{𝑏𝑖} which is a contradiction by Theorem 4.14(vi). If |𝐡𝑖|=β„΅0 and β„Ž(π‘₯)=π‘π‘–π‘›π‘œ for some π‘›π‘œ>𝑁, so all π‘π‘˜π‘–,𝑏𝑖2,…,𝑏𝑖𝑛 are in β„Ž(π‘₯). As |β„Ž(π‘₯)|=|π‘₯|>𝑁, which is impossible, so π‘₯ is an infinite set.

Recall that in ACDH-1 space, if 𝐾 is a countable dense subset of 𝑋, then by Theorem 4.14 there are countable subsets 𝐴1,𝐴2,… of 𝐾 satisfying (i)–(vii) of the pointed theorem.

Moreover, ⋃𝐴=π‘šπ‘–=1π΄π‘–βŠ†πΎ, 𝐴=𝐾=𝑋. Therefore, there is β„Žβˆˆπ»(𝑋) such that β„Ž(𝐴⧡𝐹)=𝐾⧡𝐹. By Proposition 4.15, 𝐾⧡𝐹 can be decomposed in the same way as 𝐴⧡𝐹.

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

Theorem 4.17. Let (𝑋,𝜏) be an ACDH-1 space, K any countable dense subset of 𝑋, and 𝐹 a related finite set. Then ⋃𝐾⧡𝐹=βˆžπ‘–=1𝐴𝑖 and |𝐴𝑖|=1.

Proof. Let 𝐼={π‘–βˆΆ|𝐴𝑖|=β„΅0}. If πΌβ‰ βˆ…, then 𝐴𝑖={π‘Žπ‘–1,π‘Žπ‘–2,…}, for some π‘–βˆˆπΌ. We have π‘Žπ‘–1∈Cl{π‘Žπ‘–2} and {π‘Žπ‘–1,π‘Žπ‘–2}∩𝐹=βˆ…, so, by Theorem 4.16, we have Cl{π‘Žπ‘–1} which is an infinite set. Let π‘Žπ‘–0βˆ‰πΉ with π‘Žπ‘–0∈Cl{π‘Žπ‘–1}⧡{π‘Žπ‘–1}. In a similar way, let π‘Žπ‘–βˆ’1βˆ‰πΉ with π‘Žπ‘–βˆ’1∈Cl{π‘Žπ‘–0}⧡{π‘Žπ‘–0}. By the same argument, we have a sequence …,π‘Žπ‘–βˆ’π‘›π‘Žπ‘–βˆ’π‘›+1,…,π‘Žπ‘–βˆ’1,π‘Žπ‘–0 and π‘Žπ‘–βˆ’π‘˜βˆˆCl{π‘Žπ‘–βˆ’π‘˜+1}⧡{π‘Žπ‘–βˆ’π‘˜+1}. Now we claim that for all π‘˜,π‘›βˆˆβ„•,π‘Žπ‘–βˆ’π‘˜β‰ π‘Žπ‘–π‘›. If π‘Žπ‘–βˆ’π‘˜=π‘Žπ‘–π‘› for some 𝑛β‰₯2 and π‘˜β‰₯0, then π‘Žπ‘–π‘›βˆˆCl{π‘Žπ‘–1} which contradicts the fact that {π‘Žπ‘–1,π‘Žπ‘–2,…} is a closurely ordered set. Also π‘Žπ‘–1β‰ π‘Žπ‘–βˆ’π‘˜ for all π‘˜>0, since π‘Žπ‘–1βˆ‰Cl{π‘Žπ‘–0} and π‘Žπ‘–βˆ’π‘˜βˆˆCl{π‘Žπ‘–0}, so we proved our claim. Let 𝐾1⋃=[π‘–βˆ‰πΌπ΄π‘–β‹ƒ]βˆͺ[π‘–βˆˆπΌ{…,π‘Žπ‘–βˆ’2π‘Žπ‘–βˆ’1,π‘Žπ‘–0,π‘Žπ‘–1,π‘Žπ‘–1,…}]. Then 𝑋=πΎβŠ†πΎ1βˆͺ𝐹, hence 𝐾1βˆͺ𝐹 is a countable dense subset of 𝑋. Then there is β„Žβˆˆπ»(𝑋) such that β„Ž(𝐾⧡𝐹)=(𝐾1βˆͺ𝐹)⧡𝐹=𝐾1. For π‘–βˆ‰πΌ,𝐴𝑖={π‘Žπ‘–} and π‘Žπ‘–βˆ‰π‘₯, for all π‘₯∈(𝐾⧡𝐹)⧡{π‘Žπ‘–}. Then by Proposition 4.15, β„Ž(𝐴𝑖)={π‘Žπ‘—}=𝐴𝑗, π‘—βˆ‰πΌ. Now define 𝑖0=inf(𝐼), therefore 𝐴𝑖0={π‘Žπ‘–01,π‘Žπ‘–02,…}. Moreover, β„Ž(π‘Žπ‘–01)βˆˆπ΄π‘—0, for some π‘—π‘œβˆˆπΌ, where 𝐴𝑗0={…,π‘Žπ‘—0βˆ’2,π‘Žπ‘—0βˆ’1,β‹―,π‘Žπ‘—01,π‘Žπ‘—02,…}. Then β„Ž(π‘Žπ‘–01)=π‘Žπ‘—0π‘˜, for some π‘˜βˆˆβ„€. Also π‘₯∈Cl{π‘Žπ‘–01}, where π‘₯∈𝐾⧡𝐹 and π‘₯=β„Žβˆ’1(π‘Žπ‘—0π‘˜βˆ’1). If π‘₯βˆˆπ΄π‘Ÿ, for some π‘ŸβˆˆπΌ, then π΄π‘Ÿβ‹ƒβˆ©Cl{π‘Ÿβˆ’1𝑗=1𝐴𝑗}β‰ βˆ… which is a contradiction, so 𝐼=πœ™.

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

Corollary 4.18. If (𝑋,𝜏) is an ACDH-1 space, 𝐹 is a related finite set, and 𝐾 is a countable dense subset of 𝑋, then 𝐾⧡𝐹 has the closurely ordered property.

Proof. If 𝐾 is a countable dense subset of 𝑋, then by using Theorem 4.17, we have that ⋃𝐾⧡𝐹=βˆžπ‘–=1𝐴𝑖 and |𝐴𝑖|=1, for all π‘–βˆˆβ„•. Therefore,⋃𝐾⧡𝐹=βˆžπ‘–=1π‘Žπ‘– indeed, π‘Žπ‘–βˆ‰Cl{π‘Žπ‘—}, for all 𝑖≠𝑗.

Corollary 4.19. Every ACDH-1 space is an 𝐴𝑆𝑇1 space.

Proof. Let 𝐹 be a related finite set and π‘₯≠𝑦 with {π‘₯,𝑦}∩𝐹=βˆ…. If 𝐾 is a countable dense subset of 𝑋, say 𝐾={π‘₯1,π‘₯2,…}, then the set 𝐾1={π‘₯,𝑦,π‘₯1,π‘₯2,…} is also a countable dense subset of 𝑋, therefore by Corollary 4.18, we have that 𝐾1⧡𝐹 has the closurely ordered property and {π‘₯,𝑦}βŠ†πΎ1⧡𝐹, therefore π‘¦βˆ‰π‘₯. Similarly, π‘₯βˆ‰π‘¦.

Acknowledgment

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

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Copyright © 2010 Abdalla Tallafha and Ahmed Al-Rawashdeh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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