Question

find the limit points and closure of z and Q in R.best regards

Answer 1

#5737 Find the limit points and closure of $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$ Solution

1. It is obvious that $\mathbb{Z}$ is closed, due to $\mathbb{Z}^c = \cup_{i\in \mathbb{Z}}(i,i + 1)$. Hence closure of $\mathbb{Z}$ is $\mathbb{Z}$, and the set of limit points is empty, because each point on the real line has neighborhood that does not intersect with $\mathbb{Z}$.

2. The closure of $\mathbb{Q}$ is $\mathbb{R}$, because for each point $r \in \mathbb{R}$ exists such sequence $(q_n)_{n \geq 0} \subset \mathbb{Q}$, that $q_n \to r$, $n \to \infty$. And the set of limit points is also $\mathbb{R}$, because every neighborhood of each point of the real line intersects with $\mathbb{Q}$ infinitely many times.

3. $\mathbb{R}$ is closed, hence the closure is $\mathbb{R}$ and due to similar reasons as above the set of limit points is $\mathbb{R}$.