Is $\{\langle x,y\rangle\mid 1 \leq x \leq 2, y = 0\}$ compact in $\Bbb R^2$?

Question

Is this set in $\Bbb R^2$ compact:

$$\{\langle x,y\rangle\mid 1 \leq x \leq 2, y = 0\}$$

I think it is compact, but the answer says not. Any help is appreciated.

Answer 1

It is compact. In $\Bbb R^2$, $K$ is compact if and only if $K$ is closed and bounded. This set is just the line segment from $(1,0)$ to $(2,0)$ in the plane.

Answer 2

HINT:

Note that this is the continuous image of $[1,2]\subseteq\Bbb R$ which is compact into $\Bbb R^2$, where the map is $x\mapsto\langle x,0\rangle$.