Let $X$ be complete linear metric space. Is it true that if we remove from a dense subset $A$ of $X$ a subset which has cardinality less then cardinality of $A$ then we obtain dense subset of $X$ ? If not, what about Banach spaces?
No. Consider $X=\mathbb R$ and $A=\mathbb Q\cup [0,1]$, and remove from $A$ the subset $\mathbb Q$.
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