# examples of nowhere dense sets

Note that $\mathbb{Z}$ is nowhere dense in $\mathbb{R}$ under the usual topology: $\operatorname{int}\overline{\mathbb{Z}}=\operatorname{int}\mathbb{Z}=\emptyset$. Similarly, $\frac{1}{n}\mathbb{Z}$ is nowhere dense for every $n\in\mathbb{Z}$ with $n>0$.

This result provides an alternative way to prove that $\mathbb{Q}$ is meager in $\mathbb{R}$ under the usual topology, since $\displaystyle\mathbb{Q}=\bigcup_{n\in\mathbb{Z}\text{ and }n>0}\textstyle{% \frac{1}{n}}\mathbb{Z}$.

Title examples of nowhere dense sets ExamplesOfNowhereDenseSets 2013-03-22 17:07:05 2013-03-22 17:07:05 Wkbj79 (1863) Wkbj79 (1863) 4 Wkbj79 (1863) Example msc 54A99 ExampleOfAMeagerSet