# Balls connected in R^n

• April 8th 2009, 03:51 PM
Andreamet
Balls connected in R^n
Show that the open balls and closed balls in the standard metric on $\mathbb{R}^n$ are connected in $\mathbb{R}^n$
• April 8th 2009, 04:18 PM
Jhevon
Quote:

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

note that $S \subseteq \mathbb{R}^n$ 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