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# handout3 - Useful Theorems on open and closed sets MAA 4224...

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Useful Theorems on open and closed sets – MAA 4224 The following is a list of definitions and theorem from class. To use this list effectively, don’t just read the list. See if you can state the definitions without looking. Rewrite the proofs yourself, making sure that you follow the logic. There may be some typos. See if you can find any. Definition. A is open if x A, > 0 , V ( x ) A. Definition. x R is a limit point of A if > 0 , V ( x ) A \ { x } 6 = . The set of limit points is denoted L A . Definition. x A is an isolated point of A if > 0 , V ( x ) A = { x } . Definition. A is closed if L A A . Theorem 1 Any point in A that is not an isolated point of A is a limit point of A . Proof. Let x A . x is not an isolated point of A = > 0 , ¬ [ { x } = V ( x ) A ] = > 0 , V ( x ) A \ { x } 6 = contains an element other than x . Theorem 2 A is closed ⇐⇒ A c is open. 1

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Proof. ( ) Take x A c . Want to show: > 0 , V ( x ) A c .
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