Please solve numbers #1,#2,#3, and #4.

1. a. Prove that a topological space X is Hausdorff if and only if the set A = {(1, x): I E X } is a closed subset of X x X. (One direction of "if and only if" suffices.) b. Consider the statement: If f: X -> Y and for every A C X, f(A) Cf(A), then f is continuous. (i) State the contrapositive of this statement. (ii) Prove this statement (or its contrapositive). 2. a. Prove that if (X, d) is a metric space then the usual e - o characterization is equivalent to the open set definition of continuity. (One direction of "is finer" suffices.) b. If Ti is a strictly finer topology than 72 on the same set X, what can you say about closures of subsets in either topology? Justify your answer. c. Find the closure of the set (-2, -1) U (1, 2) in the lower limit topology. Justify your answer. 3. Find the closure of the subset A = {(x, I): I E [0, 1] \\ Q} of the ordered square 12 = [0, 1] x [0, 1] (in the dictionary order topology). Justify your answer. Bonus. If possible find a sequence in the set A that converges to a corner of the square 12. 4. a. Determine whether the function f: R > Rw. defined by f (t) = (t, jt, it, }t, it,..) = (2 "t). is continuous when Ro is equipped with either the product or the box topology. b. Consider the sequence (an) , C R" defined by 1 = (1, 0, 0, 0, 0, 0, ... ) 12 = (0, , 0, 0, 0, 0, ...) 1 = (0, 0, , 0, 0, 0, ...) if n = k else @1 = (0, 0, 0, , 0, 0, . .. ) Determine in which of the (i) product, (ii) box, and (iii) uniform topologies the sequence (a.) converges. Briefly justify your answers