Interior of a set in $\mathbb{R}^2$. I am studying Munkres topology on my own.
I want to find  the interior of each of the following subsets of $\mathbb{R}^2$.
$$A = \{x \times y \mid y = 0\}$$
To find the interior, I want show that if $x \times 0$ be an arbitrary point of $A$, then there does not exists any neighborhood $U$ of $x \times 0$ such that $U \subset A$. And to do that I want to show $U$ be a nbd of $x \times 0$ such that $U \subset A$, then it is a contradiction. Hence $\text{Int } A = \emptyset$.
But I can not execute the above idea. Please help me.

If $(x,0)\in A$, and if $r>0$, then $\left(x,\frac r2\right)\in D_r\bigl((x,0)\bigr)$. But $\left(x,\frac r2\right)\notin A$. So, every open disk centered at $(x,0)$ contains points outside $A$, and therefore $(x,0)\notin\mathring A$. Since this occurs for each element of $A$, $\mathring A=\emptyset$.
