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 Further properties of 1-sequence-covering maps
Tác giả hoặc Nhóm tác giả: TRAN VAN AN AND LUONG QUOC TUYEN
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Nơi đăng: Đại hội Toán học Toàn quốc lần thứ 7 tại Quy Nhơn
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; Số: 07;Từ->đến trang: 208;Năm: 2008
Lĩnh vực: Tự nhiên; Loại: Báo cáo; Thể loại: Trong nước
TÓM TẮT
In 1996, Lin [4] gave the notion of 1-sequence-covering maps and proved that a space is an 1-sequence-covering and s-image of a metric space if and only if it has a point-countable sn-network, and a space is an 1-sequence-covering, quotient and s-image of a metric space if and only if it has a point-countable weak base, and then Lin and Yan proved that every sequence-covering, quotient and s-image of a locally separable metric space is a local ℵ0-space in [5]. Recently, Xia [7] introduced the concept of weak-open maps, and by using it, certain gf-countable spaces are characterized as images of metric spaces under various weak-open maps. π-map is an another important map which was
introduced by Ponomarev in 1960, and π-images of metric spaces cause attention once again in [2], [3], [6].
The purpose of this paper is to establish some relationships between 1-sequence-covering maps and weak-open maps or sequence-covering s-maps, and also to give a generalization of a result in [5]. The our main results are the following
Proposition 0.1. Let f : X −→ Y be a sequence-covering map, and Y be snf-countable. If (1) or (2) satisfies, then f is an 1-sequence-covering map.
(1) f is an s-map, and X has a point-countable base;
(2) f is a Lindel¨of map, and X is first countable.
Corollary 0.2. Let f : X −→ Y be a map. If one of the following conditions satisfies,
then f is an 1-sequence-covering map.
(1) f is a sequence-covering s-map, X has a point-countable base, and Y is gf-
countable;
(2) f is a sequence-covering Lindel¨of map, X is first countable, and Y is gf-
countable;
(3) f is a weak-open map, and X is first countable.
Theorem 0.3. If f : X −→ Y is a sequence-covering π-s-map, then f is an 1-sequence-
covering map.
Corollary 0.4. Let f : M −→ X be a map. If M is a metric space, then the following are equivalent.
(1) f is a weak-open π-s-map;
(2) f is an 1-sequence-covering, quotient π-s-map;
(3) f is a sequence-covering, quotient π-s-map.

References

[1] A. V. Arhangel’skii, Mappings and spaces, Russian Math. Surveys, 21 (4) (1966), 115-162.
[2] Y. Ge, Spaces with countable sn-networks, Comment. Math. Univ. Carolinae, 45 (1) (2004), 169-176.
[3] Z. Li, On π-s-images of metric spaces. Int. J. Math. Math. Sci., 7 (2005), 1101-1107.
[4] S. Lin, On sequence-covering s-mappings, Adv. Math. (China), 25 (6) (1996), 548-551.
[5] S. Lin and P. Yan, Sequence-covering maps of metric spaces, Topology Appl., 109 (2001), 301-314.
[6] Y. Tanaka and Y. Ge, Around quotient compact images of metric spaces, and symmetric spaces,
Houston J. Math., 32 (1) (2006), 99-117.
[7] S. Xia, Characterizations of certain g-first countable spaces, Adv. Math., 29 (2000), 61-64.

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