Which points in the digital line form open sets which form

Question

Q1. Let X be a set and C be a collection of subsets of X whose union is all of X.

(a) Let B_C be the collection of all subsets of X that can be expressed as an intersection of finitely many of the sets from C. Show that B_C is a basis for a topology on X.

(b) Let T_C be the topology generated by the basis BC. The collection C is called a subbasis for the topology T_C. Check (prove) that C ⊆ T, and so every set in C is an open set in the topology T_C.

(c) Let T be a topology on X containing C. Prove that TC ⊆ T. Conclude that T_C is the smallest topology on X containing C.

Q2. Consider the set Z of integers. Define sets B(n) as follows:

(a) Show that the collection B = {B(n) | n ∈ Z} is a basis for a topology on Z. (Z with this topology is called the digital line.)

(b) Show that the digital line is not the same as Z with the discrete topology.

(c) Which points in the digital line form open sets? Which form closed sets? For the points which are not closed, find the smallest closed set containing that point.

Q3. Repeat the above exercise for the set Z × Z, this time with sets B(m, n) defined as follows:

The set Z × Z with this topology is called the digital plane.