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Mathematics
Topology and Analysis
Proving a function f is continuous given A U B = X
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[QUOTE="mathwonk, post: 6554169, member: 13785"] By one definition of continuity, it suffices show that the inverse image C of an open set of Y under f, is open in X. By hypothesis its restriction, i.e. intersection, with both A and B are open. So we want to show that if X = AunionB, and both A and B are open, and if CmeetA is open in A, and CmeetB is open in B, then C is itself open in X. Fortunately, since A and B are open in X, being open in A or B is the same as being open in X. Then use the formula C = CmeetX = Cmeet(AunionB) = (CmeetA) union (CmeetB). Similarly, continuity of f is equivalent to showing that the inverse image D of a closed set in Y, is closed in X. We are given that DmeetA and Dmeet B, are closed in A and B respectively, Again, being closed in a closed subset of X is the same as being closed in X... If it is any consolation to you, presumably as a newbie, I admit that even as a seasoned old professional, I found this tedious and had to use pen and paper. However, as an aid to future problems, continuity is a "local" property, which means if it is true on each set of an open cover, then it is true. So the case where A and B are open, is "obvious" to someone who knows that. It had not occurred to me that it is also true for A and B closed, and presumably this is because we need the closed cover to be finite, i.e. we need to use the fact that a finite union of closed sets is closed, whereas any union of open sets is open. i.e. I would guess that the correct generalization of this problem is to show that if f is continuous when restricted to each set of any open cover of X, then f is continuous on X, and if f is continuous on each set of any finite closed cover of X, then f is continuous on X. You might try showing these when you finish the current problem. This would be more useful later on than just the result stated in the problem. [/QUOTE]
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Proving a function f is continuous given A U B = X
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