Determine if $X = \{(x,y) \in \mathbb R^2 \mid x^2 + x^3y^2 = 0\}$ is compact in $(\mathbb R^2, \mathcal E_2)$, where $\mathcal E_2$ denotes the standard Euclidean topology.
I know that $X$ is closed, being the set of roots of a continuous function, and that $X = \{(x,y) \in \mathbb R^2 \mid x = 0 \lor x = -1/y^2\}$ from the definition of $X$. Since $X$ contains the real line, which is not compact in $\mathcal E_1$, I'm inclined to believe that neither $X$ is compact, but I don't know how to prove it formally.
A subset of $\Bbb R^2$ is compact if and only if it is closed and bounded (Heine–Borel theorem). Here $X$ is not bounded since $(0,y)\in X$ for all $y\in\Bbb R$.