1) If f is a continuous mapping from a matric space X to metric space Y. A E is a subset of X.
The prove that f(closure(E)) subset of closure of f(E).
2) Give an example where f(closure (E)) is a proper subset of closure of f(E).
The Attempt at a Solution
My problem is the second part, although I am unsure of my solution to part (1) as well.
If E is closed then E= closure of E. Hence, for all x that belong to E , f(x) belong to Y. Hence, f(closure of E) = f(E) = set of all f(x). Now, the closure of the set of all f(x) shall be f(x) (as collection of f(x) in Y is also closed).
Hence, the closure of f(E)= f(closure of (E)).
Now, if E is open, then closure of E = E U E', where E' contains limit points. Now, f(closure of E) = f(E) and the set of limit points that are not in E.
Where as f(E) shall be f(x) .
(Kind of lost here).
Also, can't think of a possible example that satisfies (2). I was thinking f(x) = 1/x , where x = (0,infinity). But this does not follow proper subset example.