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chels124
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Homework Statement
Let X and Y be metric spaces, f a function from X to Y:
a) If X is a union of open sets Ui on each of which f is continuous prove that f is continuous on X.
b) If X is a finite union of closed sets F1, F2, ... , Fn on each of which f is continuous, prove that f is continuous on X.
c) Can part (b) be extended to an infinite number of closed subsets?
Homework Equations
The following three are equivalent:
(i) f is continuous
(ii) The complete inverse image of an open set is open
(iii) The complete inverse image of a close set is closed.
The Attempt at a Solution
(summarized obviously)
A) Is it enough to say that we know for each open set that makes up X, there exists an inverse image of that set in Y. Since this is equivalent to continuity, we know f is continuous on X.
B) Similarly, we know for each closed set that makes up X, there exists an inverse image of that set in Y. Since this is equivalent to continuity, we know f is continuous on X.
C) This is the part that confused me and made me feel like (A) and (B) were incorrect. I saw somewhere that it cannot be extended to an infinite number of closed subsets, but I don't understand why. If we use the equivalences from above, why wouldn't the function be continuous? So, I thought maybe I summed everything up a bit too quickly, but I can't find my own contingencies.
Thanks for any help!