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Convergence of a Sequence in a Finer Topology

  1. Jan 5, 2017 #1
    1. The problem statement, all variables and given/known data
    Clearly if a sequence of points ##\{x_n\}## in some space ##X## with some topology, then the sequence will also converge when ##X## is endowed with any coarser topology. I suspect this doesn't hold for endowment of ##X## with a finer topology, since a finer topology amounts to more open sets, decreasing the likelihood of convergence. However, I would like to build a counterexample to settle this matter.

    2. Relevant equations


    3. The attempt at a solution

    First, I am having a embarrassing confusion with quantifiers. The definition of convergence I am working is the following: ##x_n## converges to ##x## if and only if for every open neighborhood ##U## of ##x##, there exists ##N \in \mathbb{N}## such that ##x_n \in U## for all ##n \ge N##. Which of the two is the proper negation:

    (1) ##x_n## does not converge to ##x## iff there exists an open neighborhood ##U## of ##x## such that ##x_n \notin U## for some ##n \ge N##

    (2) ##x_n## does not converge to ##x## iff there exists an open neighborhood ##U## of ##x## such that ##x_n \notin U## for every ##n \ge N##

    Once this is settled, my goal is to show that the sequence ##\frac{1}{n}## does not converge to ##0## in the lower limit topology (note, I don't actually know if this is the case; but I conjectured it on the basis that it is the simplest example).

    EDIT: Perhaps this is easier. Just consider the indiscrete topology on ##\mathbb{R}##. Then every number in ##\mathbb{R}## is a limit of ##\frac{1}{n}##, but the sequence ##\frac{1}{n}## in the standard topology only has ##0## as its limit.
     
  2. jcsd
  3. Jan 5, 2017 #2

    FactChecker

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    Neither of these are right.
    xn does not converge to x iff there exists an open set U containing x such that for every N>0, there exists n>N with xn not in U.
    This is a good example to work with (sort of trivial, but that is ok for an example). Find a sequence that doesn't converge in the standard topology and show that it converges in the indiscrete topology.
     
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