Is $\{(x,y) \in \mathbb{R}^2 \mid x^3+y^3=1 \}$ compact?

Question

Let $A$ be the subset of points $(x,y) \in \mathbb{R} \times \mathbb{R}$ such that $x^3+y^3=1$. Is $A$ compact?

I think it should be, as $x$, $y$ can't have values greater than $1$. So $A$ is bounded. Also it is closed. So, compact. But answer key says it is not compact.

Answer 1

I think it should be, as $x$, $y$ can't have values greater than $1$. So $A$ is bounded.

Odd powers can be negative, so this doesn't hold. For example: take $x=2$, then $y = \sqrt[3]{1-2^3}$.

Note that your reasoning would work for e.g. $x^2+y^2=1$.

Answer 2

It's not bounded. $x$ or $y$ can be anything greater than $1$. For example, if $x=10$ then $y=\sqrt[3]{1-x^3} = (-999)^{1/3}$.

Try plotting $x^3+y^3=1$ in a graphing calculator or desmos.com, etc., for visual confirmation.