This is one of a series of reports on the digital geometry of three-dimensional images, such as those produced by computed tomography. In this report we define simple surface points and simple closed surfaces, and show that any connected collection of simple surface points form a simple closed surface, thus proving a three-dimensional analog of the two-dimensional Jordan curve theorem. We also show that the converse is not a theorem (in contrast to the two-dimensional case), and discuss more complex surface types. Finally, we show that the two-dimensional analog of our definition of simple closed surface characterizes simple closed curves, but that several other characterizations of 2D curves, when extended to 3D, are not adequate to characterize surfaces.