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