Results 1 to 11 of 11
Like Tree2Thanks
  • 1 Post By Plato
  • 1 Post By Plato

Math Help - A counter-example?

  1. #1
    Member ModusPonens's Avatar
    Joined
    Aug 2010
    Posts
    125
    Thanks
    14

    A counter-example?

    Hello

    I'm working on the exercises of Munkres' Topology. Section 30, exercise 1-a says: "Show that in a first-countable T_1 space, every one-point set is a G_{ \delta} set"

    Now, I've seen the solution to this exercise and I believe it's wrong, because it assumes that a countable basis at x is a family of sets that get "arbitrarily small", that is, it's like a countable collection of balls of radius 1/n (I know the space doesn't need to be metrizable, I'm just giving the intuition). This assumption is wrong because there could be a strange topology in R in which the neighbourhoods of 0 would be intervals of the form (-1-1/n,1+1/n). This is a countable basis at 0.

    Now the counter example: R with the finite complement topology. In this topology, it is T_1 and first countable, but no one-point set is a G_{ \delta} set.

    Is this counter-example right? Thanks in advance
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1573
    Awards
    1

    Re: A counter-example?

    Quote Originally Posted by ModusPonens View Post
    Munkres' Topology. Section 30, exercise 1-a says: "Show that in a first-countable T_1 space, every one-point set is a G_{ \delta} set"

    Now, I've seen the solution to this exercise and I believe it's wrong, because it assumes that a countable basis at x is a family of sets that get "arbitrarily small", that is, it's like a countable collection of balls of radius 1/n (I know the space doesn't need to be metrizable, I'm just giving the intuition). This assumption is wrong because there could be a strange topology in R in which the neighbourhoods of 0 would be intervals of the form (-1-1/n,1+1/n). This is a countable basis at 0.

    Now the counter example: R with the finite complement topology. In this topology, it is T_1 and first countable, but no one-point set is a G_{ \delta} set.
    "An example of a space which is not first-countable is the cofinite topology on an uncountable set (such as the real line)."
    Thanks from ModusPonens
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member ModusPonens's Avatar
    Joined
    Aug 2010
    Posts
    125
    Thanks
    14

    Re: A counter-example?

    Thank you, but I don't understand why. Why isn't the family \{ B_n : B_0=R- \{ x+1 \} ;B_n=B_{n-1}- \{x+n \} \} a countable basis at x?

    And if you can explain that, please explain how to prove the original question.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1573
    Awards
    1

    Re: A counter-example?

    Quote Originally Posted by ModusPonens View Post
    Thank you, but I don't understand why. Why isn't the family \{ B_n : B_0=R- \{ x+1 \} ;B_n=B_{n-1}- \{x+n \} \} a countable basis at x?
    How do you define a local basis?
    And why do you think the above is one?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member ModusPonens's Avatar
    Joined
    Aug 2010
    Posts
    125
    Thanks
    14

    Re: A counter-example?

    I will transcribe the definition from the book.

    A space X is said to have a countable basis at x if there is a countable collection B of neighborhoods of x such that each neighborhood of x contains at least one of the elements of B. A space that has a countable basis at each of its points is said to satisfy the first countability axiom, or to be first-countable.

    I think that the family I defined is a countable family of neighborhoods of x, such that each neighborhood contains at least one other (in fact infinite neighborhoods). Since x is arbitrary, that would make the space first-countable.

    PS: Neighborhood is defined just as an open set containing x, and not a set which contains an open set containing x.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1573
    Awards
    1

    Re: A counter-example?

    I will simply prove that \mathbb{R} with the cofinite topology cannot be first countable. Suppose that a\in\mathbb{R} and \{O_n\} is a local basis at a
    \forall n,~\mathbb{R}\setminus O_n is finite.
    Thus [\bigcup\limits_n {\mathbb{R}\backslash {O_n}}  must be countable.
    So  \bigcap\limits_n {{O_n}}  must be uncountable and contain a point b such b\ne a .
    But Q=\mathbb{R}\setminus\{b\} is an open set and a\in Q .
    Now it is impossible for O_n\subset Q .
    Contradiction .
    Thanks from ModusPonens
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1573
    Awards
    1

    Re: A counter-example?

    Quote Originally Posted by ModusPonens View Post
    Munkres' Topology. Section 30, exercise 1-a says: "Show that in a first-countable T_1 space, every one-point set is a G_{ \delta} set"

    Now, I've seen the solution to this exercise and I believe it's wrong, because it assumes that a countable basis at x
    Of course it it assumes that a countable basis at x because that is what first countable means.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Member ModusPonens's Avatar
    Joined
    Aug 2010
    Posts
    125
    Thanks
    14

    Re: A counter-example?

    I'm not sure if you read my first post without attention or if I wasn't clear, but my contention is that a countable basis at x is not a collection of sets that gets arbitrarily small. That's why it's difficult to do the exercise.

    Help would be apreciated. No time constraints, this isn't a homework assignment.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Member ModusPonens's Avatar
    Joined
    Aug 2010
    Posts
    125
    Thanks
    14

    Re: A counter-example?

    Quote Originally Posted by Plato View Post
    I will simply prove that \mathbb{R} with the cofinite topology cannot be first countable. Suppose that a\in\mathbb{R} and \{O_n\} is a local basis at a
    \forall n,~\mathbb{R}\setminus O_n is finite.
    Thus [\bigcup\limits_n {\mathbb{R}\backslash {O_n}}  must be countable.
    So  \bigcap\limits_n {{O_n}}  must be uncountable and contain a point b such b\ne a .
    But Q=\mathbb{R}\setminus\{b\} is an open set and a\in Q .
    Now it is impossible for O_n\subset Q .
    Contradiction .
    Is it a contradiction because \{O_n\}\bigcup\{Q\} should form a countable basis and it doesn't?

    If that is so, why does there have to be another countable basis at a. One for each point is enough to make it first-countable.

    I'm sorry if I'm bothering you, but I can't get this out of my mind.
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Member ModusPonens's Avatar
    Joined
    Aug 2010
    Posts
    125
    Thanks
    14

    Re: A counter-example?

    How about this as a counterexample to the original question: \mathbb{R} with the following subbasis: \{\mathbb{R}-\{x\}:x\in\mathbb{R}\}\cup\{(k-1/n,k+1+1/n):k\in\mathbb{Z},n\in\mathbb{N},n>10\}
    Follow Math Help Forum on Facebook and Google+

  11. #11
    Member ModusPonens's Avatar
    Joined
    Aug 2010
    Posts
    125
    Thanks
    14

    Re: A counter-example?

    In this topology, the set \{1/2\} is not a G_{\delta} set.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. when is a counter example not enough?
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: September 20th 2011, 11:34 AM
  2. box counter problem
    Posted in the Statistics Forum
    Replies: 6
    Last Post: May 15th 2010, 07:54 AM
  3. counter examples!
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: December 8th 2009, 01:09 PM
  4. counter example
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: December 1st 2009, 11:07 PM
  5. I need a counter example
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: April 13th 2009, 01:34 AM

Search Tags


/mathhelpforum @mathhelpforum