Q1 Let N be the set of all positive integers. Prove that each of the following collections of subsets of N is a topology. (i) Tị consists of N, Ø, and every set {1,2, ..., n}, for n any positive...

Extracted text: Q1 Let N be the set of all positive integers. Prove that each of the following collections of subsets of N is a topology. (i) Tị consists of N, Ø, and every set {1,2, ..., n}, for n any positive integer. (This is called the initial segment topology.) (ii) T2 consists of N, Ø, and every set {n, n+1,. }, for n any positive integer. ... (This is called the final segment topology.)