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On spaces with point-countable cs-networks

✍ Scribed by Shou Lin; Chuan Liu

Publisher: Elsevier Science
Year: 1996
Tongue: English
Weight: 473 KB
Volume: 74
Category: Article
ISSN: 0166-8641
DOI: 10.1016/s0166-8641(96)00043-0

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✦ Synopsis

In this paper we discuss three questions about the quotient s-images of metric spaces. The main results are:

(1) X is a sequential space with a point-countable cs-network if and only if X is a compactcovering, sequence-covering, quotient and s-image of a metric space.

(2) Let X and Y be sequential spaces with point-countable cs-networks, then X x Y is a k-space if and only if one of the three properties below holds. (a) X and Y are first countable spaces. (b) X or Y is a locally compact space. (c) X and Y are local k~-spaces.

(3) Let f : X -+ Y be a pseudo-open s-map. If X is a Fr6chet space with a point-countable cs-network, then Y is a Fr6chet space with a point-countable cs*-network.

They partly answer three questions posed by Michael and Nagami (1973), Tanaka (1983), and Gruenhage, respectively.

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