Consider $A = \mathbb{R} \times \{0\} \subset \mathbb{R}^2$ . what is the boundary of $A$ in $\mathbb{R}^2$?

Question

Consider $A = \mathbb{R} \times \{0\} \subset \mathbb{R}^2$ . what is the boundary of $A$ in $\mathbb{R}^2$ ?

My thinking : $A = \mathbb{R} \times \{0\}$ isomorphic to $\mathbb{R}$ then boundary of $A$ is $\mathbb{R}$

Finally my answer is $\mathbb{R}$

Answer 1

$A$ is closed and it has no interior. So its boundary is $A$ itself. You cannot say that the boundary is $\mathbb R$ even though it is homeomorphic to $\mathbb R$.