Example of a dense set which is not nowhere dense

Question

I can't find an example of a dense set A in $\mathbb{R}^{2}$ which is not nowhere dense. The definition of nowhere dense set given in the exercise is as follows:

Let $\mathrm{X}$ be a topological space. A subset $\mathrm{A}$ $\subset$ $\mathrm{X}$ is nowhere dense in $\mathrm{X}$ if $\mathrm{X}$$\setminus$$\mathrm{A}$ is dense in $\mathrm{X}$.

I thought of something like $\mathbb{Q} \times (\pi\mathbb{Q})$ as I saw in this question, but I'm not sure that the complementary set is not dense in $\mathbb{R}^{2}$. Thank you for helping me.

How about $A=\mathbb R^2$? It is dense but its complement $\varnothing$ is not. — drhab, Dec 28 '19 at 15:22
Yes you're right! I was thinking only of proper subset but this is not necessary. Thank you, it was easier that I thought! — Laura Gigliotti, Dec 28 '19 at 15:26

score 1 · Accepted Answer · answered Dec 28 '19 at 15:31

The definition in your text defines a co-dense subset (the complement is dense), don't confuse it with the bona fide usual concept of a nowhere dense subset.

Take a closed set with empty interior and take its complement to get a lot of examples, e.g. in $\Bbb R^2$:

$\Bbb R^2$ itself (trivial)
$\{(x,y): y \neq 0\}$
$\{(x,y): \|x\|\neq 1\}$

and many more.

Example of a dense set which is not nowhere dense

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