Let $(X,\tau)$ be a topological space and $(x_{n})$ be a sequence in $X$. An accumulation point $x_{0}$ of the sequence $(x_{n})$ is defined as, for each neighborhood $U$ of $x_{0}$ and each $n\in\mathbb{N}$, there exists a $m\geq n$ such that $x_{m}\in U$. It is known that if this $x_{n}$ has a subsequence $(x_{n_{k}})$ converging to $x_{0}$, then $x_{0}$ is an accumulation point of $(x_{n})$. But the statement converse, that is, there need not be a subsequence converging to an accumulation point of $(x_{n})$. I am trying to find an example for that statement. Is there any basic example for that? or any example. I would greatly be appreciated.
Asked
Active
Viewed 29 times
1
-
Related: Cluster point of a sequence and limit point of some subsequence. – Anne Bauval Apr 27 '23 at 08:40
-
thank you very much. Arens Fort space is the one I could not understand. so why (0,0) is cluster point? @AnneBauval – Woodx Apr 27 '23 at 10:04
-
Because its neighborhoods are infinite. – Anne Bauval Apr 27 '23 at 11:38
-
Thank you very much @AnneBauval – Woodx Apr 27 '23 at 13:13