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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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