How to show whether the following sets are compact or not:

Question

How to show whether the following sets are compact or not:

$1.\{(x,y,z)\in \mathbb R^3:x^2+2y^2-3z^2=1\}$

$2.\{(x,y,z)\in \mathbb R^3:|x|+2|y|+3|z|\leq 1\}$

I know in order for a set to be compact it must be closed and bounded.But it is possible to check this $\Bbb R^2$ ,I can't do it for the above two sets

Answer 1

Note that both sets are closed, being inverse images of closed sets in $\mathbb{R}$ under continuous functions; the first set is the inverse image of $\{1\}$ under $(x,y,z)\mapsto x^2+2y^2-3z^2$; the second set is the inverse image of $(-\infty,1]$ under $(x,y,z)\mapsto|x|+2|y|-3|z|$.

The second set is $$ \{(x,y,z)\in\mathbb{R}^3:|x|+2|y|+3|z|\le1\} $$ and it is compact, because it's contained in the set $[-1,1]^3$; indeed, when $|x|>1$, $|y|>1$ and $|z|>1$, then $|x|+2|y|+3|z|>6$.

Answer 2

Hint: If we want to show it's not bounded (and hence not compact), it suffices to find elements of arbitrarily large norm.

For example, for the first one, we can take $x$ and $y$ both as large as we like, and then just make $z$ big enough so the equation is satisfied. This would give us a norm as large as we want.

In this question, we can interpret this graphically (in 3D): the (usual) norm is the distance away from the origin, so we're looking for points that get arbitrarily far.