# Example of a nowhere dense subset of a metric space.

We recall the definition of a nowhere dense subset of a metric space:

"A subset $A$ of a metric space $(X,d)$ is nowhere dense if $Int(\bar A)=\emptyset$"

I don't understand how it is that $\mathbb Z\subset \mathbb R$ is nowhere dense; how can the interior of its closure be the empty set?

• What is the problem? – Aloizio Macedo Aug 7 '15 at 6:59
• What's the closure of $\mathbb{Z}$? What's its interior? – Huy Aug 7 '15 at 7:01

The closure of $\mathbb{Z}$ is $\mathbb{Z}$. Thus, $\operatorname{Int}(\mathbb{Z}) =\emptyset$.

In other words, given $p \in \mathbb{Z}$, there does not exist an open set $U$ so that $p \in U \subset \mathbb{Z}$. This is because $\mathbb{Z} \subset \mathbb{Q}$, and $\mathbb{Q}$ is dense in $\mathbb{R}$. Thus, $U \cap \mathbb{R}$ contains some element of $\mathbb{R}$. So, $\mathbb{Z}$ is nowhere dense because every open subset around an integer contains a real number not in $\mathbb{Z}$.

The set $\Bbb Z$ consists entirely of isolated points, so it has no limit points. Hence, its closure is $\Bbb Z$.

Also because $\overline{\Bbb Z}=\Bbb Z$ consists of isolated points, it has no interior points, meaning its interior is empty.

To see any $k \in \Bbb Z$ is an isolated point, note that the only integer in the open interval $\left( k- \frac 12, k+\frac 12 \right)$ is $k$.

• In topology, an isolated point is one whose singleton is open. You appear to be using this term incorrectly. – user642796 Aug 7 '15 at 8:34
• @ArthurFischer OP didn't specify a definition, so I used this one: "In particular, in a Euclidean space (or in any other metric space), x is an isolated point of S, if one can find an open ball around x which contains no other points of S." (From the second paragraph on the site you quoted.) This is also the definition we used in my real analysis class. Or using the first paragraph, here $X=\Bbb R$, $S = \Bbb Z$, and the neighborhood of $X$ containing no other points of $S$ is the open interval. Could you be more specific about why I'm misusing it? – coldnumber Aug 7 '15 at 8:39