Fort Space Topology
For the Fort space topology, let be a sequence in X, such that the set of the sequence, , is infinite.
Using the definition of convergence in topological space, prove that 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 and . B is infinite and , so is a T-neighbourhood of p, and there is an , such that, for all .
Is anybody able to help with this?
Thanks in advance.
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 such that
So let V be a neighbourhood of p. We proved here that the complement of V is finite.
Hence, the subset such that is finite.
Define (it exists since it's a finite set).
Then for any , and that's all ! :)