# Showing a set is compact

1. Nov 6, 2013

### MuIotaTau

1. The problem statement, all variables and given/known data

Explain why $f(x,y,z) = x + y - z$ must attain both a maximum and a minimum on the sphere $x^2 + y^2 + z^2 = 81)$.

2. Relevant equations

None

3. The attempt at a solution

I know that any continuous function attains both a maximum and a minimum on a compact set. I defined the set $K$ as $$K = \{(x,y,z)|g(x,y,z) = 0\}$$ and demonstrated that it is bounded by a closed ball. I then realized, however, that I'm not really sure how to show that the set is closed. I understand the definition of a closed set, but I don't know how to show that it is true in this particular case. Any hints?

2. Nov 6, 2013

### pasmith

What is g?

It suffices to show that the sphere is compact. To do that, you can write down a continuous surjection from a compact subset of the plane to the sphere and use the result that the continuous image of a compact space is compact.

3. Nov 6, 2013

### Staff: Mentor

What's g(x, y, z)?

4. Nov 6, 2013

### MuIotaTau

Oops, I'm sorry, I defined $g$ as $g(x,y,z) = x^2 + y^z + z^2 - 81$. So the constraint equation.

We haven't demonstrated that compactness is continuous invariant in class, unfortunately, so I would be required to prove that. If demonstrating the set is closed is more complicated, I'll do it, but I would rather not prove a theorem if it's not strictly necessary.

5. Nov 6, 2013

### Dick

Proving that the set of points where g(x,y,z)=0 is closed should be easy. Just use the definition of continuity directly or that the inverse image of a closed set is closed for a continuous function. Now use Bolzano-Weierstrass.

Last edited: Nov 6, 2013