Which of the following are compact I need Hint...

Question

Which of the following are compact?

1. $\{(x,y) \in \mathbb{R}^2 :(x-1)^2+(y-2)^2=9\} \cup \{(x,y) \in \mathbb{R}^2: y=3\}$.

2.$\{(\frac{1}{m},\frac{1}{n}) \in \mathbb{R}^2:m,n\in \mathbb{Z}-\{0\}\} \cup \{(0,0)\} \cup \{(\frac{1}{m},0) :m\in \mathbb{Z}-\{0\}\}\cup \{(0,\frac{1}{n}) :n\in \mathbb{Z}-\{0\}\}$.

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

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

what is the easiest way to find it is compact or not ? I use the "compact sets are closed and bounded" but can't imagine the image of the above problem.

Answer 1

Well, for number 1, the set contains the line $y=3$. Could this be bounded?

Number 4 looks compact, as it is closed and bounded (why?).

Answer 2

To boost your imagination,

First option looks something like

And third option,

(second image is taken from alpha)