To prove union of connected sets is not connected.

Question

Let V be span of $(1,1,1) $and $(0,1,1)$.Let $p=(0,0,1),q=(1,1,0),r=(1,0,1)$ Then prove that $ \Bbb R ^3-V \cup$ $\{tp+(1-t)r:t \in [0,1]\}$ is not connected.
$V=\{a(1,1,1)+b(0,1,1):a,b \in \Bbb R\}$ ,Let $A=\Bbb R ^3-V$ and B=$\{tp+(1-t)r:t \in [0,1]\}$ We know that if A ,B are connected sets and $A \cup B$ is connected if $A\cap B \neq \emptyset$.My doubt is how to prove intersection of A and B is empty?

Answer 1

The set $\{tp+(1-t)r\}$ has no mutual points with $V$, because all points from $V$ have the form $(a,b,b)$ and all points from $\{tp+(1-t)r\}$ have the form $(1-t,0,1)$. So, $\{tp+(1-t)r\}$ lies in $A$ and then $A\cup \{tp+(1-t)r\}$ is simply $A$ and therefore not connected.