Thread: Balls connected in R^n

Show that the open balls and closed balls in the standard metric on $\displaystyle \mathbb{R}^n$ are connected in $\displaystyle \mathbb{R}^n$

Originally Posted by Andreamet

Show that the open balls and closed balls in the standard metric on $\displaystyle \mathbb{R}^n$ are connected in $\displaystyle \mathbb{R}^n$

note that $\displaystyle S \subseteq \mathbb{R}^n$ is path-wise connected, implies that it is connected. thus, you have the desired result, if you can show that open balls and closed balls are path-wise connected.

note that any two points in an open or closed ball can be connected by a straight line that is contained in the ball