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Math Help - Topology questions

  1. #1
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    Questions

    Q: removed
    Last edited by johndon; May 21st 2010 at 02:53 AM. Reason: correcting the question
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  2. #2
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    Quote Originally Posted by johndon View Post
    Q1: Let ℝ be given the usual topology. Find a function f: ℝ→ ℝ
    such that f is continuous at x = 0 , and f is discontinuous at all x ≠ 0.
    The "Dirichlet function" is a famous example. Let f(x)= 0 if x is irrational and f(x)= 1/n where x= m/n reduced to lowest terms.

    Q2: Define a metric d on ℝ so that the sequence converges to
    1.
    What sequence?

    Q3: Define a topology τ on ℝ so that the sequence Xn=1/n converges
    to 0 , and also converges to 1, but does not converge to any other number. Is
    τ metrizable?
    Take T to consist of all open sets of R (under the usual topology) except that "1" is added to any set containing "0" and "0" is added to any set containing "1".
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  3. #3
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    missing sequence

    the sequence is Xn=1/n in Q2
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Here is for the second part of question 3

    Is this space metrizable?
    Hint:
    Spoiler:

    Theorem: Let X be Hausdorff, then if \{x_n\}_{n\in\mathbb{N}} is a convergent sequence in X, then it's limit x is unique.

    Proof: Suppose that x\ne y and let U_x,U_y be the respective disjoint neighborhoods of x,y guaranteed by Hausdorffness. Now, since x_n\to x all but finitely elements of K=\{x_n:n\in\mathbb{N}\} lie in U_x and so not all but finitely man values of K may lie in U_y. It follows that x_n\not\to y. The conclusion follows.

    Soo..................


    Quote Originally Posted by johndon View Post
    the sequence is Xn=1/n in Q2
    Work? Where is it?
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