how do i prove this $S=\{(x,y,z): x^2 +4y^2 +4 z^2 -2x +16y +40z +113 <0\}$ is an open set

Question

Elipsoide in the three-dimensional space, with the graph of the ellipsoid the set is open, but how do I prove it?

Let $f:X \to Y$ be continuous. If $U$ is an open subset of $Y$, then $f^{-1}(U)$ is an open subset of $X$. Does this help? — ThePortakal, Oct 16 '17 at 19:25
But what is $f^{-1}((-\infty, 0))$ and how does one show that is open? — fleablood, Oct 16 '17 at 19:36
@fleablood. The topological definition of continuous is inverse images of open sets are open. — William Elliot, Oct 16 '17 at 19:54
Brainfart. Idiotically I was have trouble getting that $f^{-1}((-\infty,0)) = S$. Don't know why I was haven't the glitch.... — fleablood, Oct 16 '17 at 20:21

score 1 · Answer 1 · answered Oct 16 '17 at 19:25

1

If $f(x,y,z)$ is a polynomial of three variables, then $f$ is continuous. Since $f$ is continuous, what can we say about $f^{-1}(U)$ when $U\subset\Bbb R$ is open?

answered Oct 16 '17 at 19:25

Dave

13,568

how do i prove this $S=\{(x,y,z): x^2 +4y^2 +4 z^2 -2x +16y +40z +113 <0\}$ is an open set

1 Answers1