New Types of Almost Countable Dense Homogeneous Space

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.