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 $\displaystyle \{A_n\}_{n=1}^{\infty}$ be a sequence of path connected subsets of a space X such that for each integer n>=1, $\displaystyle A_n$ has at least one point in common with one of the preceding sets $\displaystyle A_1, ...,A_{n-1}$. Then, $\displaystyle \bigcup_{n=1}^{\infty}A_n$ is path connected.
Let T be a digital topology on $\displaystyle \mathbb{Z}$. Without loss of generality, let x, y be arbitrary pair of points $\displaystyle (x \leq y )$ in $\displaystyle \mathbb{Z}$.
Choose $\displaystyle A_1$ be {x} if x is odd, or {x-1, x, x+1} if x is even. Since every basis element of $\displaystyle (\mathbb{Z}, T)$ is path connected (verify this), if y belongs to $\displaystyle A_1$, then we are done. Otherwise choose $\displaystyle A_2$ be {x, x+1, x+2} if x is odd, or {x+1} if x is even. This process continues until we find $\displaystyle A_{y-x+1}$ that contains y. Since each $\displaystyle A_n$ and $\displaystyle A_{n+1}$, n=1, 2, ... ,y-x, has an intersection and is path connected, we conclude that $\displaystyle (\mathbb{Z}, T)$ is path connected by lemma 1.