New types of almost countable dense homogeneous space (Q978990)

Summary: In [Fundam. Math. 74, 189--194 (1972; Zbl 0207.21102)], \textit{R. Bennett} studied countable dense homogeneous (CDH) spaces and in [Topology Appl. 44, No.~1--3, 143--147 (1992; Zbl 0767.54013)], \textit{B. Fitzpatrick jun., J. M. S. White} and \textit{H. Zhou} proved that every CDH space is a \(T_{1}\) space. Afterward \textit{A. Al-Bsoul, A. Fora} and \textit{A. Tallafha} [Almost countable dense homogeneous spaces, to appear] gave another proof for the same result, also they defined almost CDH spaces and almost \(T_{1}, T_{0}\) spaces, indeed they proved that every ACDH space is an almost \(T_{1}\) space. In this paper we introduce a new type of almost CDH spaces called ACDH-1, we characterize ACDH spaces and almost \(T_{0}\) spaces, and we also give relations between different types of CDH spaces. We define new types of almost \(T_{1} (AT_{1})\) 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 \(AT_{1}\).