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Unformatted text preview: Chapter 3: Basic Topology of R PWhite Discussion Open and Closed Sets Compact Sets Chapter 3: Basic Topology of R Peter W. White [email protected] Initial development by Keith E. Emmert Department of Mathematics Tarleton State University Fall 2011 / Real Anaylsis I Chapter 3: Basic Topology of R PWhite Discussion Open and Closed Sets Compact Sets Overview Discussion: The Cantor Set Open and Closed Sets Compact Sets Chapter 3: Basic Topology of R PWhite Discussion Open and Closed Sets Compact Sets Cantor’s Monster Definition 1 I Begin with C = [ , 1 ] . I Let C 1 = C \ ( 1 3 , 2 3 ) . I Let C 2 = C 1 \ ( 1 9 , 2 9 ) ∪ ( 7 9 , 8 9 ) . I Let C n be the set obtained by removing all the “middle thirds” from C n 1 , for n ≥ 1. I Then, C = ∞ \ n = C n . The resulting set C is called the Cantor set . Chapter 3: Basic Topology of R PWhite Discussion Open and Closed Sets Compact Sets Cantor’s Monster 1 1 3 2 3 1 9 2 9 7 9 8 9 C C 1 C 2 C 3 1 27 2 27 7 27 8 27 19 27 19 27 25 27 26 27 . . . . . . . . . . . . . . . . . . . . . . . . . . . I C 6 = ∅ since 1 ∈ C n for all n . In fact, if x is an endpoint of C n , then x ∈ C . I We have removed intervals of length 1 3 , 2 · 1 9 , 4 · 1 27 ,... : 1 3 + 2 · 1 9 + 4 · 1 27 + ··· + 2 n 1 · 1 3 n + ··· = 1 . Thus, the Cantor set has zero length. Chapter 3: Basic Topology of R PWhite Discussion Open and Closed Sets Compact Sets Fractional Dimension I The dimension of a point is zero. I The dimension of a line segment is one. I The dimension of a square is two. I The dimension of a cube is three. I Scaling: If we magnify the size of the above creatures by three, we obtain I One = 3 copies of the point. I Three = 3 1 copies of the line segment. I Nine = 3 2 copies of the square. I Twentyseven 3 3 copies of the cube....
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This note was uploaded on 01/17/2012 for the course MATH 409 taught by Professor Dr.peterwhite during the Fall '11 term at Tarleton.
 Fall '11
 Dr.PeterWhite
 Topology, Sets

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