Geometry Problem: Does a Plane Exist Where p Non-Intersecting Lines Meet?

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Homework Statement


In R^3, if we have p non-intersecting lines, is it true that there exists a plane Y such that all the lines intersect Y? Up to rotations.

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The Attempt at a Solution



No real attempt, this is just something I found to be interesting. Any thoughts?
 

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Here is a better question:

Is R^3 - {n non-intersecting lines} homeomorphic to R^3 - {n parallel non intersecting lines}?
 

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