lec9 - Math 117 – April 26, 2011 9 Topology of the reals...

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Unformatted text preview: 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 ( ∃ > ,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....
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This note was uploaded on 05/31/2011 for the course MATH 117 taught by Professor Akhmedov,a during the Spring '08 term at UCSB.

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