How to show that $M:=\{(a,b,c)\in\Bbb R^3 \,\,∣\,\,a^3+b^3+c^3−3abc=1\}$ is not a compact space?

Question

How to show that $M:=\{(a,b,c)\in\Bbb R^3 \,\,∣\,\,a^3+b^3+c^3−3abc=1\}$ is not a compact space? I know that for proving that space is not compact, it must be not closed or not bounded or (both) not closed and not bounded, but I want a hint how to show this?

Answer 1

since $a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$ so let's choose $a\to 0, b=-c$, we have $a(a^2+3b^2)=1$ or $3b^2=\frac1a-a^2$, so b is unbounded when $a\to 0$