  According to a general definition of discrete curves, surfaces, and
manifolds. This paper focuses on the Jordan curve theorem in 2D discrete
spaces. The Jordan curve theorem says that a (simply) closed curve separates a
simply connected surface into two components. Based on the definition of
discrete surfaces, we give three reasonable definitions of simply connected
spaces. Theoretically, these three definition shall be equivalent. We have
proved the Jordan curve theorem under the third definition of simply connected
spaces. The Jordan theorem shows the relationship among an object, its
boundary, and its outside area.
  After the publication of the first version of the paper ({\it L. Chen, Note
on the discrete Jordan Curve Theorem. In: SPIE Conf. on Vision Geometry VIII,
vol. 3811, pp. 82-94. (1999).}), we found some statements in the original proof
of the Jordan Curve Theorem were not explained well. One case was not proven in
details. In this revision, we added two more minor definitions and make the
proof more solid and sound when it is needed for embedding a discrete surface
into a Euclidean space.
  In this revision, we also proved that the third definition of simply
connected spaces equivalent to the second definition of simply connected
spaces.
