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Problem : What must be true of a surface in order for it to be a simple closed surface?

The surface must divide space into three distinct regions: the surface itself, the interior of the surface, and the exterior of the surface.

Problem : If a line is perpendicular to a plane, is that line perpendicular to every line in the plane?

No. The line is only perpendicular to every line in the plane that contains the intersection point of the first line and the plane.

Problem : If a polyhedron has 6 faces, how many edges does it have?

There is not enough information to know this. The answer depends on how many sides each face has.

Problem : Is a surface two-dimensional or three-dimensional?

A surface itself is two-dimensional: it has no thickness. A surface can, however, span three dimensions. A polyhedron does not exist is a single plane--it spans three dimensions, but the surface itself is still two-dimensional.

Problem : Is it possible for a surface to be contained in a single curve?

Generally speaking, no. Surfaces are two-dimensional and curves are one-dimensional, so this is impossible. Consider the following situation, though: Curve One is a line segment of length 10. Curve Two is a line segment of length 3. Curve two moves only within the line that contains it. Thus, the surface that traces the motion of curve two is actually a line segment. Its length depends on how far Curve two moves. It is possible for the surface of the motion of Curve two to be contained in Curve one, whose length is greater than that of Curve two. So in this sense, yes, it is possible. But such a surface isn't really a surface. It is like a curve that is actually a point because the curve traces the motion of a motionless point. The situation is rather obscure and useless. Still, these ideas are interesting to ponder.