Is any topology on a countable set first countable?

Question

Is any topology on a countable set first countable?

I think the answer is yes, but I just want a clarification of my reasoning:

Let $X$ be any countable set. First, I would like to recall the fact that any countable set with discrete topology is indeed second countable because it has a countable base $\{\{x\}\mid x\in X\}$.

We know that if a topological space is second countable, then it remains second countable under a coarser topology (i.e., if $\tau_1$ and $\tau_2$ are two topologies on $X$ such that $\tau_1\subseteq\tau_2$ then $(X,\tau_1)$ is second countable whenever $(X,\tau_2)$ is second countable).

Now any topology $\tau$ on $X$ is indeed coarser than the discrete topology, and hence $X$ with any topology is second countable. But any second countable topological space is first countable, and hence $X$ with any topology is first countable.

Is this reasoning correct?

https://math.stackexchange.com/questions/tagged/solution-verification: For posts looking for feedback or verification of a proposed solution. "Is my proof correct?" is too broad or missing context. Instead, the question must identify precisely which step in the proof is in doubt, and why so. — Anne Bauval, Mar 24 '25 at 10:30
This question is similar to: Is every countable space first countable?. If you believe it’s different, please [edit] the question, make it clear how it’s different and/or how the answers on that question are not helpful for your problem. — Bruno Andrades, Mar 24 '25 at 10:31
See https://math.stackexchange.com/questions/133898/is-every-countable-space-first-countable — Bruno Andrades, Mar 24 '25 at 10:31
And https://math.stackexchange.com/questions/3272545/are-countable-topological-spaces-second-countable#:~:text=On%20the%20other%20hand%20both,then%20it%20is%20second%2Dcountable. — Bruno Andrades, Mar 24 '25 at 10:31
@Ibrahim The statement "We know that if a topological space is second countable, then it remains second countable under a coarser topology" is false. This is where the argument breaks down. — Dean Miller, Mar 24 '25 at 11:23
Questions like these can be answered by π-Base: π-Base, Search for Countable+~First countable — Steven Clontz, Mar 29 '25 at 01:58

Is any topology on a countable set first countable?

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