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Math Help - Fort Space Topology

  1. #1
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    Fort Space Topology

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

    Using the definition of convergence in topological space, prove that (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 S=\{a_{n}:n\in\mathbb{N}\}and B=S\sim\{p\}. B is infinite and p\notin B, so \sim B is a T-neighbourhood of p, and there is an N\in\mathbb{N}, such that, for all n>N, ~a_{n}\in\sim B.
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  2. #2
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    Is anybody able to help with this?

    Thanks in advance.
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  3. #3
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    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 N\in\mathbb{N} such that 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 \mathcal{N} \subset \mathbb{N} such that n\in\mathcal{N} \Rightarrow a_n \not\in V is finite.

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

    Then for any n>N,a_n\in V, and that's all !
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