The Jordan curve theorem is one of those frustrating results in topology: it is intuitively clear but quite hard to prove. In this note we will look at two discrete analogs of the Jordan curve theorem that are easy to prove by an induction argument coupled with some geometric intuition. One of the surprises is that when we discretize the plane we get two Jordan curve theorems rather than one, a consequence of the interplay between two natural products in the category of graphs. Topology in this context has been studied by Farmer in . To state the discrete versions, we need to know what the discrete analog of the plane is and what plays the role of a simple closed curve. Since the plane is the topological product of two lines, we take as our discrete analog the product of two discrete lines. We will use undirected graphs for our analogs of spaces, with vertices for points and edges connecting points which are to be thought of as touching.
Stout, Lawrence, "Two Discrete Forms of the Jordan Curve Theorem" (1988). Scholarship. 92.