A Limit Point that is not a Sequence Limit

Last updated Nov 1, 2022

Consider $\mathbb{R}$ equipped with the Cocountable Topology. Consider $S = \mathbb{R} \setminus {0}$. Then $0$ is a Limit Point of $S$. To see this, observe ${0}$ cannot be Open because ${0}^{C}$ is Uncountable. Thus any $U \subset \mathbb{R}$ Open must be s.t. $U \cap S \neq \emptyset$.

However, $0$ cannot be a sequence limit for any Sequence on $S$. Indeed, let $({x}{n}){n=1}^{\infty} \subset S$. Then consider the Open set $U = {0} \cup {x_n: n \in \mathbb{N}}^{C}$. $U$ is indeed Open becase $U^{C} \subset {x_n}{n=1}^{\infty}$. On the other hand, since $0 \not\in {x_n}{n=1}^{\infty}$, there is no $n \in \mathbb{N}$ for which $x_{n} \in U$. Thus $x_{n} \not\to 0$. $\blacksquare$