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In 1972, Bennett studied the countable dense homogeneous (CDH) spaces
and in 1992, Fitzpatrick, White, and Zhou proved that every CDH
space is a

In 1920, Sierpinski introduced in [

In 1974, Lauer defined in [

In the first part of this paper, we introduce the definitions given by Fora et al. in [

In Section

Finally, the following abbreviations and symbols will be used throughout this paper. For a subset

Fora et al. in [

A space

In the previous definition the condition

If X is countable and

Let

A topological space

Clearly every

A topological space

A topological space

Every ACDH space is

Let

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

A topological space

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

If

Let

As an application of Proposition

If

Let

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

Let

Now let prove the following characterizations of ACDH spaces.

If

For any two countable dense subsets

For any two countable dense subsets

For any two countable dense subsets

(i) implies (ii) Suppose that

(iv) implies (i) Let

Let

Consequently, we have the following result.

Every ACDH-1 space is an ACDH space.

Let

Let

Conversely, assume that

Consequently, we have the following result.

Every ACDH-1 space is SACDH.

A transposition on

A space

A space

Let

The converse is obvious by choosing

One can show that the spaces

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

Let

If

If

Let

Now we have the following result.

If

Suppose that

Consequently, we have the following Corollary.

If

We know that almost CDH space is not a

Let

Assume that

A space

One may easily prove the following proposition.

Let

In the following results we show that the new separation axiom

Every

Suppose that

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

Let

Therefore, for all

One may easily prove the following proposition.

Every

Fitzpatrick et al. proved in [

Let

A countable collection

Let

Let

if

having closurely ordered property, is a topological property,

every subset of a set having closurely ordered property must have closurely ordered property.

Now let us define the following.

A countable set

If

The first part is clear. To prove the converse, assume that there is such a finite set

Every ACDH-1 space is

Assume that

If

each

each

if

if

It is easy to prove the following result.

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.

Let

Suppose that

Recall that in ACDH-1 space, if

Moreover,

The following theorem shows that all the above

Let

Let

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

If

If

Every ACDH-1 space is an

Let

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