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Chapter3

# Chapter3 - Chapter 3 Basic Topology of R PWhite Discussion...

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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

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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 0 = [ 0 , 1 ] . I Let C 1 = C 0 \ ( 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 = 0 C n . The resulting set C is called the Cantor set .

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Chapter 3: Basic Topology of R PWhite Discussion Open and Closed Sets Compact Sets Cantor’s Monster 0 1 1 3 2 3 1 9 2 9 7 9 8 9 C 0 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 0 copies of the point. I Three = 3 1 copies of the line segment. I Nine = 3 2 copies of the square. I Twenty-seven 3 3 copies of the cube. I In the above cases, dimension is an exponent of the magnification. So, it is reasonable to say that the number of copies equals the magnification raised to the dimension.

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Chapter 3: Basic Topology of R PWhite Discussion Open and Closed Sets Compact Sets Fractional Dimension Now, we magnify the Cantor set by three.
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