Results 1 to 5 of 5

Math Help - Discrete topology on an uncountable set

  1. #1
    Junior Member
    Joined
    Oct 2009
    Posts
    72

    Discrete topology on an uncountable set

    Prove that the discrete topology on an uncountable set does not satisfy the second axiom of countability.

    The hint says: It is not enough to prove that one basis is uncountable but every basis is uncountable.

    I can however only think of one possible basis and it's uncountable: \{\{x\}|x\in X\}

    Is the hint misleading me?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Oct 2009
    Posts
    72
    As soon as I typed it out I thought of other baseses (what's the plural for this)?

    Every basis though has to contain that basis so it will be uncountable. Is that right?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by magus View Post
    As soon as I typed it out I thought of other baseses (what's the plural for this)?

    Every basis though has to contain that basis so it will be uncountable. Is that right?
    Bases.


    The easy proof is that if it were second countable then it would be separable and so have a countable dense subset but (evidently?) the only dense subset is the full space which is uncountable.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Oct 2009
    Posts
    72
    I'm not sure what separable means and I don't think I can use that.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by magus View Post
    I'm not sure what separable means and I don't think I can use that.
    It means countable dense set and I bet you can.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Uncountable set with cofinite Topology
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: September 28th 2011, 03:35 PM
  2. Order topology = discrete topology on a set?
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: August 6th 2011, 11:19 AM
  3. Discrete Topology
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: March 12th 2011, 12:10 PM
  4. discrete topology, product topology
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: December 13th 2008, 02:19 PM
  5. discrete topology, product topology
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: December 7th 2008, 01:01 PM

/mathhelpforum @mathhelpforum