Assignment:
a) Let M be a connected topological space and let f : M ---> R be continuous. Pick m1,m2 2 M and suppose that f(m1) < f(m2). Let x 2 R be such that f(m1) < x < f(m2). Show that there is m M with f(m) = x. (Hint: Use a connectedness argument.)
b) Give R1 the usual product topology as the product of infinite copies of the real axis. Let f : R1 --> R1 be given by f(x1, x2, x3, x4, x5, . . . ) = (x1, x3, x5, . . . ). Show that f is continuous.
Provide complete and step by step solution for the question and show calculations and use formulas.