basic-defns

basic-defns - Basic Definitions Let S ⊂ Rn . and p ∈...

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Unformatted text preview: Basic Definitions Let S ⊂ Rn . and p ∈ Rn . • S is bounded if it is contained in some ball in Rn . • S is a neighborhood of p if S contains some open ball around P . • A point p is a limit point of S if every neighborhood of p contains a point q ∈ S , where q = p . • If p ∈ S is not a limit point of S , then it is called an isolated point of S . • S is closed if every limit point of S is a point of S . • A point p ∈ S is an interior point of S if S contains a neighborhood of p . • S is open if every point of S is an interior point of S . 1 • Let S denote all of the limit points of S . ¯ Then the closure S of S is the set S ∪ S . It is the smallest closed set containing S and is thus the intersection of all the closed sets containing S . • A subset T ⊂ S is dense in S if every point of S is either in T or a limit point of T (or both). R EMARK : These definitions are unchanged for any metric space instead of just Rn . 2 ...
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This note was uploaded on 03/06/2012 for the course MATH 508 taught by Professor Staff during the Fall '10 term at UPenn.

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basic-defns - Basic Definitions Let S ⊂ Rn . and p ∈...

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