If $\mathbb R^3\setminus V$ connected where $V$ is the subspace generated by $\{(1,1,1),(0,1,1)\} $

linear-algebra vector-spaces connectedness

I believe it's good to think about this problem geoemetrically. $V$ is a plane in $\mathbb{R}^3$, to so $\left(\mathbb{R}^3\setminus V\right) \cup S$ will be connected if and only if $S\cap V\neq\emptyset$. Can you see why?

With this in mind, the problem boils down to figuring out which of the sets $S$ intersect $V$ in each of the four cases.

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