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Math Help - Real Number Analysis: About Metric Spaces

  1. #1
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    Real Number Analysis: About Metric Spaces

    I never imagined that there would be a day where I would need to ask for help on mathematics, but sadly (for me anyway), today happens to be the day.

    Enough, I'll cut to the chase. I have no idea how to do these problems:

    Homework #3
    -----------

    All problems assume that the set E is a subset of a metric space \left ( X,d \right ).

    The following problems only require what was covered through Thursday's lecture:

    1. Show that each neighborhood in a metric space is convex.
    (check text for definition).
    2. Prove that E' is closed.
    3. Prove that the derived sets of E and of the closure of E coincide.
    4. Do E and E' have the same set of limit points?
    5. Prove that the closure of E is a closed set.

    6. Construct a bounded set of real numbers with exactly three limit points.
    7. Let X be an infinite set. For p an element of X and q an element of X, define d \left ( p,q \right ) = 1 if p \not= q, and d \left( p,q \right ) = 0 if \left (p = 1 \right ). Prove that \left ( X,d \right ) is a metric. Which subsets of the resulting metric space are open? Which are closed? Which are compact?

    And definitions:

    • Convex: We call a set E contained in R^k convex if:

      \lambda x + \left (1 - \lambda \right ) y \in E

      when x \in E, y \in E and 0 < \lambda < 1.
    • E' is the set of limit points of E.
    • A point p is a limit point of the set E if every neighborhood of p contains a point q \not= p such that q \in E.
    • A neighborhood of a point p is a set N_r \left ( p \right ) consisting of all points q such that d \left ( p, q \right ) < r. The number r is called the radius of N_r \left ( p \right ).



    I blame the text (Principles of Mathematical Analysis, Rudin 1976) and the professor for the opacity of lessons.

    If anyone can walk me through, or provide a quick guide, I'd be very, very grateful!
    Last edited by RNAS; October 6th 2008 at 08:25 PM. Reason: Fixed laTeX error
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  2. #2
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    Can I get this thread moved to the Calculus section? I just noticed that the calculus section includes real analysis.
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