Find the boundary and interior of $\mathbb R \times \mathbb N$ in $\mathbb R^2$

Question

I need to find the boundary and the interior of $$\mathbb R \times \mathbb N$$ I visualized this set as a bunch of horizontal lines given by the equation $y = n$ where $n \in \mathbb N$. Given this picture, I can infer that the interior of this set is $\emptyset$ (It is not possible to create any circle contained in this set). Also, as for the boundary, every point on each of these lines can be limit of a sequence contained in $\mathbb R \times \mathbb N$ therefore I would say that the set itself is the boundary.

Is my answer to the problem correct? If not, what am I missing?

Answer 1

You answer is correct. Be careful though, the boundary is not the set of points which are limits of sequences whithin the subspace: any point x is limit of the constant sequence $x_n=x \ \forall n$. I rather say that if $X$ is your subspace and $\overset{\circ}{X}$ is its interior, then the boundary is defined as $\partial X := \overline{X} - \overset{\circ}{X}$. But in your case $X=\mathbb{R} \times \mathbb{N}$, you just need to prove that $X$ is closed so you get $\partial X = \overline{X} - \emptyset = X$