Connected subsets of $\mathbb{R^3}$

Question

Question: Let $V$ be span of $(1,1,1)$ and $(0,1,1)∈\mathbb{R^3}$, let $u_1=(0,0,1)$, $u_2=(1,1,0)$, $u_3=(1,0,1)$ 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.

$3.(\mathbb R^3\backslash V)\cup \{ tu_1+(1-t)u_3:0\le t\le 1$} is connected.

$4.(\mathbb R^3\backslash V)\cup \{(t,2t,2t):t\in \mathbb R$} is connected.

My attempt: I know what is definition of connected sets, but usually I had applied the definition to subsets of $\mathbb{R}$ but yet i didn't applied it subsets of bigger space $\mathbb{R^3}$. But to start the problem, first I had to find $V$.

Now, $V= span\{(1,1,1),(0,1,1)\}$

$= \{x(1,1,1)+y(0,1,1): x, y ∈\mathbb{R^3}\}$

$=\{(x,x+y,x+y): x,y ∈\mathbb{R}\}$

Now, what will be, $(\mathbb R^3\backslash V)$? it's complement of $V$ in bigger space $\mathbb {R^3}$, beyond my imagination!! Got stuck! From hours, Please help me..

Answer 1

1. $V$ is a plane in $\mathbb{R}^3$. If you would consider $\mathbb{R}^3\setminus V$, then you would have divided $\mathbb{R}^3$ into two halves. However, you have added the origin back, so you have two open, connected sets sharing a point, making the first one a connected space.

For the other three sets $\mathbb{R}^3\setminus V \cup A$ (with $A$ the sets defined in your problem), try to find out if these sets intersect $V$ or not. If they do intersect $V$, the space becomes connected by the same reasoning as in $(1)$. If the sets do not intersect $V$, then you have a disconnected set.

EDIT: your question is exactly the same as the this one, which has been answered apparently.

SECOND EDIT: Consider (for example) $\mathbb{R}^3$ with the $xy$-plane removed, i.e. $\mathbb{R}^3\setminus \{(x,y,0) \vert x,y \in \mathbb{R}\}$. The space you obtain from this operation is $$\{(x,y,z) \vert x,y \in \mathbb{R}, z > 0\} \cup \{(x,y,z) \vert x,y \in \mathbb{R}, z < 0\}$$ both of them being open sets in $\mathbb{R}^3$ (and hence also in $\mathbb{R}^3$ with the $xy$-plane removed), they form a separation of the space (a separation of a space $X$ is a disjoint union of open subset $U,V$). Note that the space $\mathbb{R}^3\setminus V$ is homeomorphic (read "the same") to the space $\mathbb{R}^3$ with the $xy$-plane removed.

You also asked me to do one of the last three. Let me give a 'hint' for $(3)$ and $(4)$: what happens if you set $t = 0$?