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Counterexample Topology

  1. Dec 7, 2008 #1
    1. The problem statement, all variables and given/known data

    Let [tex]( X, \tau_x)[/tex] [tex] (Y, \tau_y)[/tex] topological spaces, [itex](x_n)[/itex] an inheritance that converges at [tex]x \in X[/tex], and let [tex]f_*:X\rightarrow Y[/itex].
    Then, [tex]f[/itex] is continuos, if given [itex](x_n)[/itex] that converges at [tex]x \in X [/tex], then [tex]f((x_n))[/itex] converges at [tex]f(x)[/itex].
    I need a counter example, to prove that the reciprocal is not true.

    All I know is that X should not be first countable.
    Please, help me.

    Thanks in advance.
    Last edited: Dec 7, 2008
  2. jcsd
  3. Dec 8, 2008 #2


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    I think there are some translation problems here. [itex](x_n)[/itex] is a "sequence" not an "inheritance". And you want to show that the "converse" of that statement, not the "reciprocal", is false.

    The converse of "If for any sequence [itex](x_n)[/itex] that converges to x, [itex](f(x_n))[/itex] converges to f(x) then f is continuous at x" is "if f is continuous at x, then for any sequence [itex](x_n)[/itex] that converges x, [itex](f(x_n))[/itex] converges to f(x)".

    I wonder if you don't have the statement reversed. The converse, as stated, IS true and there is no counter example.

    However, if the original statement were "if f is continuous at x and [itex](x_n)[/itex] is a sequence that converges to x, then [itex](f(x_n))[/itex] converges to f(x)", its converse, "if [itex](x_n)[/itex] is a sequence converging to x and [itex](f(x_n))[/itex] converges to f(x), then f is continuous at x" is false. It might happen that there exist such a sequence (but other sequences,[itex](a_n)[/itex] converging to x for which [itex](f(a_n))[/itex] does NOT converge to f(x)) but f(x) is not continuous at x.

    To look for a counter example, an obvious thing to do is to look at functions that are NOT continous at some number x in the real line. Giving different formulas to rational and irrational x might be useful.
  4. Dec 8, 2008 #3
    That`s true.

    Thanks for help.
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