Assignment:
Q1. a) Suppose T_1 is a topology on X = {a,b,c} containing {a}, {b} but not {c}. Write down all the subsets of X which you know are definitely in T_1. Be careful not to name subsets which may or may not be in T_1.
b) Suppose T_2 is a topology on Y = {a,b,c,d,e} containing {a,b}, {b,c}, {c,d} and {d,e}. Write down all the subsets of X which you know are definitely in T_2. Be careful not to name subsets which may or may not be in T_2.
c) Invent a topology T_3 on Z = {a,b,c,d} containing {a}, {b,c} and {a,d}, but not {d}
Q2. What is the only topology you can have on the one point set P = {x_0}? Describe it explicitly. Suppose (X, T) is a topological space. What is the only function you can have f: X-->P? Is it always sometimes or never continuous? Justify your answer.
Q3. Let X be a topological space with at least two points and the indiscreet topology. Why is this not a topology that comes from a metric on X? Hint: you might like to try a proof by contradiction.
Provide complete and step by step solution for the question and show calculations and use formulas.