Take any four points in space. Draw all lines connecting pairs of them. Then draw all lines connecting pairs of points on those lines.

Can the resulting set of points cover all of space?

The six lines connecting the four points cannot be parallel. Take any point in space. If the point is on one of the lines, the problem is solved. If it isn't, pick any of the six lines, and consider the plane determined by the line and the point. At least one of the other lines will intersect the plane. Join that intersection point with the original point, and intersect this new line with the original line. Solved!