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Unformatted text preview: 3.2 Open sets and closed sets 47 3.2 Open sets and closed sets 3.2.1 Open sets QUESTION: Why are open sets handy? ANSWER: They have wiggle room. Definition 3.2.1. lim x n = L iff > , N, n N = | x n- L | < . This is equivalent in R to a more general definition: U (open interval containing L ) , N such that n N = x n U. REASON: L ( a,b ) = | L- a | , | L- b | > 0. Take to be the smaller of the two. Then | x n- L | < = x n ( a,b ). Other direction is similar. Definition 3.2.2. A set U is open iff every point of U lies in an open interval which is contained in U , i.e., if x U = a,b such that x ( a,b ) U. This means open sets are automatically big: they contain uncountably many points; contain EVERY point between the inf and sup of any subinterval. An open set automat- ically has positive length. Interpretation of defn: Roughly, no point of an open set lies on the boundary....
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- Fall '11