I have some home work about justifying why some sets are compact, connected or arcwise-connected. The sets are: $C=\{(x,y)\in \mathbb{R}^2 \mid x\in (0,1)\}$ $D=\{(x,y)\in\mathbb{R}^2 \mid 0\leq x^2+y^2\leq 1\}$ Now I've already proved that $C$ is not compact (it's open, and by Heine-Borell must be closed to be compact in $\mathbb{R}^2$), and if I can proof that it's arcwise-connected, then is connected, and I think this is true, but I can't find the curve $\alpha : [a,b]\mapsto C$ such that, for every $x,y\in C$: a) $\alpha(a)=x$ b) $\alpha(b)=y$ c) if $t_k\mapsto t$ then $\alpha(t_k)\mapsto\alpha(t)$ d) $\alpha([a,b])\subset C$.
For $D$, I don't know even if it's compact. I could use some help here.
$C$ is convex, hence it is connected.
$D$ is closed and bounded, hence compact. Moreover it is convex, hence connected.