It should be step by step. Ur help will be highly appreciated. It is a topology in pure mathematics.

1. Answers to the questions should be fully and clearly explained QUESTION ONE a) Define carefully the following concepts An open set The power set 181 Limit point of a set in a metric space b) Given a metric space (X, d), prove that the open ball B(xo. r) is an open set in X. c) Let (X, d) be a metric space, then F a subset of X is closed if and only if F contains all its limit points. d) Using the Principle of Mathematical Induction prove that a set with n elements has 2" subsets. QUESTION TWO a) Define carefully the following concepts: 1 A Hausdorff space. An equivalence relation 111 Neighbourhood of a point in a topological space b) Prove that all metric spaces are Hausdorff spaces. c) If (X, d) is a topological space, then a non-empty subset E of X is open if and only if & is a neighbourhood of each of its points d) Let f: A - B be an onto function and x vy if f(x) = f(y'). Is this an equivalence relation? Justify your answer Address