$\mathbb R^3$ is the vector space.The subspace $V$ is spanned by $\{u_1=(1,1,1),u_2=(0,1,1)\}.$ $$u_3=(1,0,1).$$ Then which of the following are correct ?
$1.(\mathbb R^3\backslash V)\cup \{(0,0,0)\}$ is not connected.
$2.(\mathbb R^3\backslash V)\cup \{ tu_1+(1-t)u_2:0\le t\le 1\}$ is connected.
$3.(\mathbb R^3\backslash V)\cup \{ tu_1+tu_3:0\le t\le 1$} is connected.
$4.(\mathbb R^3\backslash V)\cup \{(t,2t,2t):t\in \mathbb R$} is connected.
Now I think $V$ is like $\mathbb R^2$ and $(\mathbb R^3\backslash V)$ is not connected.But including the point $0$ makes it connected again. In fact I think all the given sets would be connected.The first set adding $(0,0,0)$ to the set $(\mathbb R^3\backslash V)$ and thus connecting it. Now the second and third one each adds a straight line to $(\mathbb R^3\backslash V)\cup \{(0,0,0)\}$ and thus connected. Option $3$ adds to $(\mathbb R^3\backslash V)\cup \{(0,0,0)\}$ anothe connected set which is the continuous image of the set $x=t$ of $\mathbb R$ under the map $f(t)=(t,2t,2t).$ So all are connected sets.
Am I correct? Thank you.