Welcome to our community

Be a part of something great, join today!

how to prove that convex set is connected

ianchenmu

Member
Feb 3, 2013
74
A set $E⊂\mathbb{R}^{n}$
is convex if, given any two points $x$,$y$$∈E$, the line segment {$x + t(y-x)$ s.t. $t∈[0,1]$}
is contained in $E$. Prove that if $E$ is convex, then $ E$ is connected.
 

Fernando Revilla

Well-known member
MHB Math Helper
Jan 29, 2012
661
A set $E⊂\mathbb{R}^{n}$ is convex if, given any two points $x$,$y$$∈E$, the line segment {$x + t(y-x)$ s.t. $t∈[0,1]$}
is contained in $E$. Prove that if $E$ is convex, then $ E$ is connected.
Even more, $E$ is pathwise connected. If $x$ and $y$ are in $E$ then, the line segment $x+t(y-x): t \in [0,1]$ is a subset of $E$, so $\gamma:[0,1]\to E,\; \gamma (t)=x+t(y-x)$ is continuous path and connects $x$ to $y$.