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I know that in 2d space, a line $L$ separates the plane into two disjoint nonempty portions, called half-planes, such that two points lie $P$ and $Q$ lie on the same half-plane iff the segment $PQ$ does not intersect $L$, and lie on opposite sides iff the segment $PQ$ does intersect $L$. How does one prove that the corresponding property for lines in 3d space is false? That is, how does one prove that no line separates 3d space into two disjoint nonempty portions, such that two points $P$ and $Q$ lie on the same portion iff the segment $PQ$ does not intersect $L$, and on different portions iff the segment $PQ$ does intersect $L$?

user107952
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  • Check this: https://math.stackexchange.com/q/2026842/42969 – Martin R Jun 05 '25 at 12:41
  • Context matters in such questions: Are you identifying Euclidean space with $\mathbb R^3$ and work with the topological notion of separation/connectivity? Or maybe you work with some axiomatic definition of Euclidean space. Then you should explain which one and what notion of separation you are using. – Moishe Kohan Jun 05 '25 at 13:37
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    If you remove a line from $\mathbb{R}^3$ it is still path connected, therefore it's connected. You can find a path explicitly between any two points by considering the cases when the path does and doesn't intersect the line. – CyclotomicField Jun 05 '25 at 15:16

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