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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 \(AT_{1}\).
54D10 Lower separation axioms (\(T_0\)–\(T_3\), etc.)
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[1] W. Sierpinski, “Sur une propiete topologique des ensembles denombrable dense en Soi,” Fundamenta Mathematicae, vol. 1, pp. 11-28, 1920.
[2] M. FrĂ©chet, “Les dimensions d’un ensemble abstrait,” Mathematische Annalen, vol. 68, no. 2, pp. 145-168, 1910. · JFM 41.0102.03 · doi:10.1007/BF01474158
[3] L. Brouwer, “Some remarks on the coherence type n,” Proceedings Akadamie van Wetenschappen Amsterdam, vol. 15, no. 2, pp. 1256-1263, 1913.
[4] R. Bennett, “Countable dense homogeneous spaces,” Fundamenta Mathematicae, vol. 74, no. 3, pp. 189-194, 1972. · Zbl 0227.54020 · eudml:214404
[5] N. Lauer, Countable dense homogeneous spaces and densely homogeneous spaces, Ph.D. thesis, Auburn University, Auburn, Ala, USA, 1974.
[6] B. Fitzpatrick Jr., J. M. S. White, and H. X. Zhou, “Homogeneity and \sigma -discrete sets,” Topology and Its Applications, vol. 44, no. 1-3, pp. 143-147, 1992. · Zbl 0767.54013 · doi:10.1016/0166-8641(92)90086-F
[7] A. Tallafha, A. Al-Bsoul, and A. Fora, “Closurely ordered countable sets and applications,” Global Journal of Pure and Applied Mathematics, vol. 1, no. 2, pp. 177-182, 2005. · Zbl 1098.54018 · www.ripublication.com
[8] A. Fora, A. Tallafha, and A. Al-Bsoul, “Almost countable dense homogeneous spaces,” to appear. · Zbl 0949.54042 · doi:10.1007/BF01474158
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