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 On a problem of Shou Lin
Tác giả hoặc Nhóm tác giả: TRAN VAN AN AND LUONG QUOC TUYEN
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Nơi đăng: Kỷ yếu Hội thảo Khoa học Kỷ niệm Nửa thế kỷ Trường Đại học Vinh Anh hùng
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; Số: Tập 1;Từ->đến trang: 19;Năm: 2009
Lĩnh vực: Tự nhiên; Loại: Báo cáo; Thể loại: Trong nước
TÓM TẮT
In 1960, P. S. Alexandroff introduced the notion of point-regulars. Then, it is a nice result that a space X is an open and compact image of a metric space if and only if X is a metacompact developable space [4], if and only if X has a point-regular base [2]. In recent years, some characterizations for certain quotient compact images of metric spaces have been obtained by means of σ-strong networks ([5], [6], [7], [9]), and the following question was posed by Ikeda, Liu and Tanaka in [6]: For a sequential space X with a point-regular cs*-network, characterize X by means of a nice image of a metric space? It is easy to see that every sequence-covering and compact image of a metric space has a point-regular cs*-network. It was proved in [10] that a space X is a sequence-covering and compact image of a metric space if and only if X has a σ-point-finite strong network consisting of cs*-covers of X. In [8], S. Lin posed the following question.
Question (Question 4, [8]). Is every Hausdorff space with a point-regular cs*--network a sequence-covering and compact image of a metric space?
In this paper, we give an affirmative answer to the Question above. As an application of this result, we give an affirmative answer to the problem posed by Y. Ikeda, C. Liu and Y. Tanaka in [6].

References
[1] P. S. Alexandroff, On the metrisation of topologycal spaces, Bull. Acad. Polon. Sci. Sér. Sci.Math. Astronom. Phys. 8 (1960), 135-140.
[2] A. V. Arhangel'skii, On mappings of metric spaces, Dokl. Akad. Nauk SSSR 145 (1962), 245-247, Translation: Soviet Math. Dokl., 3 (1962) 953-956.
[3] T. V. An and L. Q. Tuyen, Further properties of 1-sequence-covering maps, Comment. Math. Univ. Carolin., 49 (3) (2008), 477-484.
[4] R. W. Heath, Screenability, pointwise paracompactness, and metrization of Moore spaces, Canad. J. Math., 16 (1964), 763-770.
[5] Y. Ge, $\pi$-images of a metric spaces, Acta Mathematica APN., 22 (2006), 209-215.
[6] Y. Ikeda, C. Liu and Y. Tanaka, Quotient compact images of metric spaces, and related meters, Topology and its Applications, 122 (1-2) (2002), 237-252.
[7] Z. Li, S. Lin and P. Yan, A note on g-developable spaces, Far East J. Math. Sci., (FJMS) 15(2) (2004), 181-191.
[8] S. Lin, Some Problems on generalized metrizble spaces, Open Problems in Topology II-Edited by E. Peart (2007).
[9] Y. Tanaka and Y. Ge, Around quotient compact images of metric spaces, and symmetric spaces, Houston J. Math., 32 (1) (2006), 99-17.
[10] P. F. Yan, Compact images of metric spaces, J. Math. Study, 30 (1997), no. 2, 185-187.
ABSTRACT
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