Results 1 to 3 of 3

Thread: Fort Space Topology

  1. #1
    Member
    Joined
    May 2008
    Posts
    140

    Fort Space Topology

    For the Fort space topology, let $\displaystyle (a_{n}) $be a sequence in X, such that the set of the sequence, $\displaystyle \{a_{n}:n\in\mathbb{N}\}$, is infinite.

    Using the definition of convergence in topological space, prove that $\displaystyle (a_{n})$ has a subsequence which converges to p.

    I'm struggling to make the transition from a sequence to a subsequence here.

    I thought about writing $\displaystyle S=\{a_{n}:n\in\mathbb{N}\}$and $\displaystyle B=S\sim\{p\}$. B is infinite and $\displaystyle p\notin B$, so $\displaystyle \sim B $is a T-neighbourhood of p, and there is an $\displaystyle N\in\mathbb{N}$, such that, for all $\displaystyle n>N, ~a_{n}\in\sim B$.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    May 2008
    Posts
    140
    Is anybody able to help with this?

    Thanks in advance.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    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,

    Sorry, I didn't have time to answer properly this ^^'
    Anyway, the topological definition of convergence gives :

    For any neighbourhood V of p, there exists $\displaystyle N\in\mathbb{N}$ such that $\displaystyle a_n\in V~,~ \forall n>N$

    So let V be a neighbourhood of p. We proved here that the complement of V is finite.

    Hence, the subset $\displaystyle \mathcal{N} \subset \mathbb{N}$ such that $\displaystyle n\in\mathcal{N} \Rightarrow a_n \not\in V$ is finite.

    Define $\displaystyle N=\max\{n\in\mathcal{N}\}$ (it exists since it's a finite set).

    Then for any $\displaystyle n>N,a_n\in V$, and that's all !
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Fort Space Topology
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: Feb 8th 2010, 09:46 AM
  2. Topology of a metric space
    Posted in the Differential Geometry Forum
    Replies: 7
    Last Post: Feb 23rd 2009, 09:48 AM
  3. Topology Quotient Space
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: May 17th 2008, 01:42 PM
  4. Hausdorff Space- Topology
    Posted in the Calculus Forum
    Replies: 0
    Last Post: Apr 5th 2008, 05:42 PM
  5. Topology in Complex Space?
    Posted in the Calculus Forum
    Replies: 3
    Last Post: Sep 10th 2007, 04:52 PM

Search tags for this page

Click on a term to search for related topics.

Search Tags


/mathhelpforum @mathhelpforum