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What is the topological characteristic of a separable metric space X$X$ such that ...$|kX\setminus X|=\frak{c}$ for any completion $kX$ of $X$?

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What is the topological characteristic of a separable metric space X such that ...?

What is the topological characteristic of a separable metric space $X$ such that $|kX\setminus X|=\frak{c}$ for any completion $kX$ of $X$?