{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info icon This preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 2
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 .
Image of page 3

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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.
Image of page 4
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.
Image of page 5

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Chapter 3: Basic Topology of R PWhite Discussion Open and Closed Sets Compact Sets Fractional Dimension Now, we magnify the Cantor set by three.
Image of page 6
Image of page 7
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern