# Fort Space Topology

• Feb 16th 2010, 12:53 PM
Cairo
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 $n>N, ~a_{n}\in\sim B$.
• Feb 17th 2010, 09:39 AM
Cairo
Is anybody able to help with this?

• Feb 19th 2010, 11:47 AM
Moo
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 ! :)