## Ge Ying

*Spaces with countable $sn$-networks *

Comment.Math.Univ.Carolinae 45,1 (2004) 169-176. **Abstract:**In this paper, we prove that a space $X$ is a sequentially-quotient $\pi $-image of a metric space if and only if $X$ has a point-star $sn$-network consisting of $cs^*$-covers. By this result, we prove that a space $X$ is a sequentially-quotient $\pi $-image of a separable metric space if and only if $X$ has a countable $sn$-network, if and only if $X$ is a sequentially-quotient compact image of a separable metric space; this answers a question raised by Shou Lin affirmatively. We also obtain some results on spaces with countable $sn$-networks.

**Keywords:** separable metric space, sequentially-quotient ($\pi $, compact) mapping, point-star $sn$-network, $cs*$-cover

**AMS Subject Classification:** Primary 54C05, 54C10; Secondary 54D65, 54E40

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