New Types of Almost Countable Dense Homogeneous Space

In 1972, Bennett studied the countable dense homogeneous (CDH) spacesand in 1992, Fitzpatrick, White, and Zhou proved that every CDHspace is a T1 space. Afterward Bsoul, Fora, and Tallafha gave anotherproof for the same result, also they defined the almost CDH spaces andalmost 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 CDHspaces called ACDH-1, we characterize the ACDH spaces, the almostT0 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 techniquesgiven by Tallafha, Bsoul, and Fora, we prove that every ACDH-1 is an AT1