Connected:

A surface is connected if, given any two points on the surface, there is a path (that stays on the surface) that connects the two points. Essentially, a surface is connected if it is in one piece, and not connected if it is in more than one piece. The pieces of a disconnected surface are called the connected components of the surface.

For example, a sphere and the torus are connected surfaces: you can get from any point to any other point by staying on the surface. On the other hand, a pair of sphere is disconnected, since you can not get from one sphere to the other while staying on the surface of the two spheres.

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