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lec9 - Math 117 9 Topology of the reals Denition 9.1 Let x...

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Math 117 – April 26, 2011 9 Topology of the reals Definition 9.1. Let x R , > 0. A neighborhood of x of size is the set N ( x, ) = { y R : | x - y | < } . A deleted neighborhood of x of size is the set N * ( x, ) = { y R : 0 < | x - y | < } . Clearly N * ( x, ) = N ( x, ) \ { x } . In our case of the set of real numbers R , N ( x, ) = ( x - , x + ), however the above definition of a neighborhood is more general and works for any metric spaces. Using the notion of a neighborhood, points in R can be classified as interior, boundary or exterior to any particular subset S R . Definition 9.2. Let S R . A point x R is called interior to S , if there is a neighborhood of x that entirely lies in S , i.e. x N S ( > 0 , N ( x, ) S ). If for every neighborhood of x N S 6 = and N ( R \ S ) 6 = , then x is called a boundary point for S , and x is called exterior to S , if there exists a neighborhood of x , such that N S = . The set of interior points is denoted by int S , and the set of boundary points is denoted by bd S .
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