Does there exist a nowhere dense subset $X$ of a metric space such that complement of $X$ is not dense?
I know $X$ cannot be closed. Any ideas for general case?
2026-04-06 08:40:39.1775464839
Example of a nowhere dense set
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If $A$ is nowhere dense, even the larger set $\overline{A}$ has a dense complement. So a fortiori $A$ has a dense complement too; a superset of a dense set is dense.
So no.