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New types of almost countable dense homogeneous space. (English) Zbl 1197.54034
Summary: In [Fundam. Math. 74, 189–194 (1972; Zbl 0207.21102)], R. Bennett studied countable dense homogeneous (CDH) spaces and in [Topology Appl. 44, No. 1–3, 143–147 (1992; Zbl 0767.54013)], B. Fitzpatrick jun., J. M. S. White and H. Zhou proved that every CDH space is a $$T_{1}$$ space. Afterward A. Al-Bsoul, A. Fora and 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 $$AT_{1}$$.
##### MSC:
 54D10 Lower separation axioms ($$T_0$$–$$T_3$$, etc.)
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##### References:
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