1. ## 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 $\displaystyle n>N, ~a_{n}\in\sim B$.

2. Is anybody able to help with this?

3. 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 !

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### fort's topology

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