Is it true that if $C$ has measure zero and is in $[0,1]$, then $[0,1]\backslash$ $C$ dense in $[0,1]$ for any general $C\subset [0,1]$?
If not, what are some counterexamples?
2026-04-02 10:30:24.1775125824
If $C$ has measure zero and is in $[0,1]$ is $[0,1]\backslash$ $C$ dense in $[0,1]$
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This is true. For $[0,1] \backslash C$ to be dense in $[0,1]$ there would have to be an element of it in every non empty open subset of $[0,1]$. We could argue by contraposition. If $[0,1] \backslash C$ was not dense in $[0,1]$ then we could find a non empty open subset of $[0,1]$ that is disjoint with $[0,1] \backslash C$. However that means that this open subset is a subset of $C$ and since any non empty open subset of $[0,1]$ does not have measure zero that would mean that $C$ does not have measure zero.