How to check if a subset is compact and a k-dimensional submanifold?

Question

I'm given the following subset S of $\mathbb{R}^4$: $$x^4 + y^4 + z^4 + t^4 = 1$$ $$x + y + z + t = 1$$

Now, given these two, how do I show this is a compact set? I've looked at a number of examples but I'm very far behind on the knowledge required for this so I'm really not sure what to do. If anyone could provide an outline of what I need to do/use to show this, it would be super helpful!

I also needed to show that this set was a 2-dimensional submanifold of $\mathbb{R}^4$. I did this by calculating the gradient of both equations and equating them with a constant, and I got 2 sets of points, but they don't satisfy both the equations. I'm not exactly sure what to do next to show that it's a submanifold so any help in that regard would be really useful as well! :(

Thanks in advance!!

Answer 1

For the Euclidean space $E=\mathbb{R}^4,$ a subset $A$ is compact iff it is closed and bounded. The first equation (of unit sphere in $E$) gives that your intersection is bounded. The second equation is of a plane in $E$ and so is closed. A closed subset of a bounded set is closed and bounded. Therefore...