Two-Piece Property:

An immersion f of a surface M into space has the two-piece property if the pre-image (by f ) of every open half-space is connected in M. In other words, a surface has the two-piece property if every plane cuts it into at most two parts.

Any convex set has the two-piece property. Non-convex sets also can have to two-piece property, as for example the torus of revolution generated by rotating a circle about an axis that does not intersect the circle.

The two-piece property turns out to be equivalent to tightness for surfaces, which gives an important geometric interpretation to tightness.

8/12/94 dpvc@geom.umn.edu -- The Geometry Center