Prove that the digital line topology is path connected.
(the D.L.T is the topology has basis elements {n} and {m-1,m, m+1}, for n odd, m even)
Definition. A space X is path connected if every pair of points of X can be joined by a path in X.
Lemma 1. Let be a sequence of path connected subsets of a space X such that for each integer n>=1, has at least one point in common with one of the preceding sets . Then, is path connected.
Let T be a digital topology on . Without loss of generality, let x, y be arbitrary pair of points in .
Choose be {x} if x is odd, or {x-1, x, x+1} if x is even. Since every basis element of is path connected (verify this), if y belongs to , then we are done. Otherwise choose be {x, x+1, x+2} if x is odd, or {x+1} if x is even. This process continues until we find that contains y. Since each and , n=1, 2, ... ,y-x, has an intersection and is path connected, we conclude that is path connected by lemma 1.