Results 1 to 5 of 5

Math Help - Fort Space Topology

  1. #1
    Member
    Joined
    May 2008
    Posts
    140

    Fort Space Topology

    Does anybody know which sets are both open and closed, and which sets are neither open nor closed for the Fort Space Topology?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6
    Hello,

    Just write down what open and closed sets are in this topology.

    Let X be the infinite set. Fix p\in X.
    Open sets A :
    1.a) p\not\in A
    1.b) or A'=X\setminus A is finite

    Closed sets B :
    2.a) p\in B (complement of a set in which p is not)
    2.b) or B is finite.

    So now try to combine...
    It's obvious that a set can't satisfy both 1.a) and 2.a), and can't satisfy both 1.b) and 2.b)

    Now is it possible for a set to satisfy 1.a) and 2.b) ? Of course ! A finite subset of X which doesn't contain p will satisfy the two conditions. Thus it's both closed and open.
    Similarly, it's possible to find a set that satisfies 1.b) and 2.a)...


    A set that is neither open nor closed ? No it's not possible, because either p is in the set, either it's not (that's the law of excluded middle ). If it is, then the set is closed, and if it's not, then the set is open !
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    May 2008
    Posts
    140
    Thanks for this Moo.

    Am I right in thinking that the neigbourhoods of the point p, are just the open sets U of p whose complement is finite?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6
    Quote Originally Posted by Cairo View Post
    Thanks for this Moo.

    Am I right in thinking that the neigbourhoods of the point p, are just the open sets U of p whose complement is finite?
    Yes, the neighbourhood U has to contain an open set, say A, which contains p. So the complement of A will be finite (since it doesn't satisfy 1.a), it has to satisfy 1.b)).
    And since A \subset U, X\setminus U \subset X\setminus A. Thus X\setminus U is finite...
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    May 2008
    Posts
    140
    After a sleepless night and much scribbling, I am still unable to prove this result for the Fort space topology.

    Let (a_n) be a sequence in X, such that the set of the sequence, {a_n : n in N} is infinite. Using the definition of convergence in topological spaces, prove that (a_n) has a subsequence which converges to p.

    If it were a metric space, I think i could do it, but as it stands I can't.

    Could anybody offer a proof please?

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Fort Space Topology
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: February 19th 2010, 11:47 AM
  2. Topology of a metric space
    Posted in the Differential Geometry Forum
    Replies: 7
    Last Post: February 23rd 2009, 10:48 AM
  3. Topology Quotient Space
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: May 17th 2008, 02:42 PM
  4. Hausdorff Space- Topology
    Posted in the Calculus Forum
    Replies: 0
    Last Post: April 5th 2008, 06:42 PM
  5. Topology in Complex Space?
    Posted in the Calculus Forum
    Replies: 3
    Last Post: September 10th 2007, 05:52 PM

Search Tags


/mathhelpforum @mathhelpforum