Question: Show that the set $C=\{(x,y)\in \mathbb{R}^2: 1\leq x^2+y^2<2\}$ is connected.
My Question: My main question is what open sets we should pick in $\mathbb{R}^2$. Once I know what open sets we should pick I will be able to solve this.
Question: Show that the set $C=\{(x,y)\in \mathbb{R}^2: 1\leq x^2+y^2<2\}$ is connected.
My Question: My main question is what open sets we should pick in $\mathbb{R}^2$. Once I know what open sets we should pick I will be able to solve this.
A standard way to show that $C$ is connected is to observe that it is path connected. Given any two points $A$ and $B$ we can, for instance, move radially from $A$ until we reach a distance from $(0,0)$ equal to that of $B$ and then, if necessary, move on a circular arc to reach $B$.
A standard theorem in general topology tells us that a path connected set is connected.
This set is path-connected, which is clear if you draw it. It might be a bit tedious but you could actually write down explicitly the path connecting any two points $(s,t)$ and $(u,v)$ by first writing them in polar coordinates, then (if $s^{2}+t^{2}>1$) walk along the ray towards the origin until you reach the circle $S^{1}=\{(x,y);x^{2}+y^{2}=1\}$, rotate along this circle until you reach the correct angle and then again - if needed - move out along the ray coming from the origin towards $(u,v)$.
It is not even needed to move towards $S^{1}$ but this saves you discussion of different cases (the cases mentioned in the post preceding this one). You can w.l.o.g. assume that $t=0$.
Alternatively, prove that $C$ is homeomorphic to $S^1 \times [0,1)$ which is a product of two connected spaces.