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

[Geometrick]

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.

Homework Equations

The Attempt at a Solution

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

 

[Geometrick]

Here is a better question:

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