please answer whole problem 2 Y is a single point Y a b wit

Y is a single point, Y = {a, b} with topology {phi, Y, {a}}, Y = {a, b} with the discrete topology. Let (X, tau_X) be a topological space; construct a bijection between tau_X and continuous maps f: X rightarrow Y, where Y = {a, b} has topology {phi, Y, {a}}. Let X and Y be topological spaces, and C(X, Y) the set of all continuous maps from X to Y. We can place a topology on C(X, Y), called the weak topology, as follows: for every x X there is a function ev_x: C(X, Y) rightarrow Y called evaluation at x which takes a continuous function f: X rightarrow Y and evaluates f at x: ev_x(f) = f(x). The weak topology on C(X, Y) is the topology generated by {ev_x}_x X. If Y is a topological space, compute the topological space C({*}, Y), i.e., compute C({*}, Y) as a set, and compute its topology.

Solution

It is the power set of all the open sets of Y. By definition continuous map should have inverse which is also open. only open set in X is a singleton set which is {*}.so any continous function shall map to this point only. hence the power set of all open sets.
It should be clear that this topology is discrete topology on Y(by definition of discrete topology)