3. 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.
The first one I think I got, connected(M) --> connected (R) open sets in f-1 implies open sets in f, and connected definition implies there exists a m. The second is a bit more tricky, I assume its an inverse open set argument.(its the intermediate value problem, so far I'm correct and various other proofs depend on how detailed the author is.)
The second part is different. I want to say that piece wise it is continuous, or that f-1*f=(x1,x2,x3...)=f-1(x1,x2,x3,....)=f(x1,x2,x3...) so f = f-1 and since we take open sets to open sets by well defined we get a continuous function.
Please see the attached file for the complete solution.
Since is continuous, then we define as , where is considered as a fixed number given in the problem. So is also continuous. From the condition, we have , then we have
The solution discusses open sets, connectivity and continuous functions.