All of the famous examples for sequentially indistinguishable topologies on a set $X$ are provided on an uncountable set $X$ (an uncountable set $X$ with discrete and cocountable topology or the $l^1(\mathbb{N})$-space with norm and weak topology, see [this](https://mathoverflow.net/questions/36379/is-a-topology-determined-by-its-convergent-sequences) thread).


A natural question arises: Does there exist on a countable set $X$ two topologies that have the same convergent sequences? (Note that the countability of $X$ does not has a large influence on the sequentiality of the topology, e.g. the Arens-Fort space is countable and not sequential.) If this is so, can such topologies be explicitely given?



Sequentially indistinguishable topologies on a countable set
There are plenty of topologies on a countable set for which all convergent sequences are eventually constant.


The most constructive example I know is the Arens-Fort space, given as example 26 in Steen and Seebach's *Counterexamples in Topology*.


This topology is constructed on the set $\mathbb{N}^2 \cup \{\infty\}$. Every point of $\mathbb{N}^2$ is isolated, and open neighborhoods of $\infty$ are defined to contain all but finitely many points from all but finitely many columns of $\mathbb{N} \times \mathbb{N}$ (it helps to draw a picture).


Obviously no nontrivial sequence converges to a point of $\mathbb{N}^2$. Suppose $(x\_n)$ is a sequence in $\mathbb{N}^2$: we will show it does not converge to $\infty$. Either (a) it has points in only finitely many columns, so some neighborhood of $\infty$ misses the whole sequence, or (b) it has points in infinitely many columns, and we can find a infinite subsequence with one point in each column, which shows that some neighborhood of $\infty$ misses infinitely many of the $x\_n$.


A little less constructively, any countable subset of the Stone-Cech compactification of the integers will have the property that every convergent sequence is eventually constant. Using this fact, you obtain $2^{2^{\aleph\_0}}$ non-homeomorphic countable spaces, none of which have any nontrivial convergent sequences.


