In $\mathbb{R}$, are these sets open? Are they closed?
- $A = \{\frac{1}{n} : n \in \mathbb{N}\}$
- $B = A \cup \{0\} $
- $[0, 1)$
My thoughts:
$A$ is not open as if we have an open ball with $r > 0$ at any point $x$ in $A$ it will contain points that are not in $A$. $A$ is not closed as the complement of $A$ is not open. That is, any open ball at $0$ will contain points from both $A$ and the complement of $A$, namely, 0 is a boundary point.
$B$ is not open as if we have an open ball with $r > 0$ at any point $x$ in $B$ it will contain points that are not in $B$. $B$ is closed as it's complement is a union of open intervals so the complement is open and hence $B$ is closed.
$[0, 1)$ is not open as an open ball with $r > 0$ at 0 will contain points not in the set. It's complement is not open so it is not closed.
How does that look, have I got these correct?