Question

The topology T  on Z generated by a basis of the form {2n -1, 2n, 2n+1} for every integer n is known as the digital topology on Z.  Prove that (Z , T) is connected.

Answer 1

Consider some subsets of Z. Let n=1, 2, 3:

Z={1,2,3}∪{3,4,5}∪{5,6,7}={1,2,3,4,5,6,7}.

So there are no gaps in this subset so the subset is connected.

The reason there are no gaps is that 2n+1=2(n+1)-1, that is, the highest element in one subset of 3 elements has the same value as the lowest element in an adjacent (next highest) subset. Therefore the union of such subsets (of which there are an infinite number) contains all the elements of the contributing subsets. There are no non-empty sets in Z. (Z, T) is connected: the topology T applied to Z is connected.