sub math problems

bluesky8UID: 2691497

I have two sub math problems.

58 If f(z) is an analytic function that maps the entire finite complex plane into the real axis, then the imaginary axis must be mapped ontoanswer: "a point"does someone know why?

64 Let S be a compact topological space, let T be a topological space, and let f be a function from S onto T. Of the following conditions on f, which is the weakest condition sufficient to ensure the compactness of T?(A) f is homeomorphism; (B) f is continuous and 1-1; (C) f is continuous; (D) f is 1-1; (E) f is boundedanswer: (C)why f is 1-1 is not the weakest sufficient condition? and does f bounded imply T is compact?

r9zhangyuUID: 2711758

58 by C-R equation, it is trivial;

64 also trivial; 1-1 just means they have the same cardinal number, say there is a 1-1 map between [0,1] and (0,1) , but [0,1] is compact, (0,1) is not.
In 64. bounded is meaningless, since T is only a TOP space, not a metric space