{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

mth536_problem_7_23c

mth536_problem_7_23c - MTH536 Chapter 7 Problem 23c Gene...

Info icon This preview shows pages 1–9. Sign up to view the full content.

View Full Document Right Arrow Icon
MTH536 Chapter 7 Problem 23c Gene Quinn – p. 1/
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 7 Problem 23c Show that total boundedness is not a topological property. – p. 2/
Image of page 2
Proof of proposition W e can estab l ish the proposition b y e xhibiting a homeomor phism from a totally bounded metr ic space to one that is not totally bounded . – p. 3/
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
Proof of proposition W e can estab l ish the proposition b y e xhibiting a homeomor phism from a totally bounded metr ic space to one that is not totally bounded . In prob lem 8, it w as estab lished that defined b y is a homeomor phism betw een and with the usual metr ic. It remains to be sho wn that is totally bounded, and is not. – p. 3/
Image of page 4
Proof that is totally bounded Let be giv en. By the axiom of Archimedes , there is an integer Let – p. 4/
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
Proof that is totally bounded Let be giv en. By the axiom of Archimedes , there is an integer Let Then w e ha v e points in spaced at equal inter v als of , so e v er y point of is within of a point in the finite set . This estab lishes that is totally bounde d. – p. 4/
Image of page 6
Proof that is not totally bounded Let be giv en and let be an y finite set of points
Image of page 7

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

View Full Document Right Arrow Icon
Image of page 8
Image of page 9
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