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

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- Apr 8th 2009, 03:51 PMAndreametBalls connected in R^n
Show that the open balls and closed balls in the standard metric on $\displaystyle \mathbb{R}^n$ are connected in $\displaystyle \mathbb{R}^n$

- Apr 8th 2009, 04:18 PMJhevon
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