In this question the OP mentioned "$N_i$-hierarchy for various countability axioms and also that the name $N_2$-space or $N_2$-property is used for second countable space.
I did not encounter this terminology before and a quick search in Google, Google Books and Google Scholar did not reveal many resources where something like this appear. (Well, I expect this post to appear soon among the results of the Google search.)
Question: Is this terminology commonly used? Or was is used in the past? Do you know any references where it is used? What are other properties in this "$N_i$-hierarchy"?
Added later: In this comment a user mentioned that they have heard this terminology in Italy. Indeed, when I try to search for "primo numerabile" "n1" or "secondo numerabile" "n2" I get some reasonably looking hits. (However, I do not speak Italian, so I am not able to understand too much more from them, but the top hits seem to be indeed about topology.)
When I tried to search for "spazio topologico" "n1" "n2" "n3" I found some results which seems to indicate that $N_3$ is used for separable space. For example, among the results was this text, which mentions $N_1$ and $N_2$ for first and second countable spaces and which says: "Spazi $N_3$ Diremo infine che uno spazio topologico $(X,\mathcal T)$ e $N_3$ se soddisfa il terzo assioma di numerabilita: esiste un sottoinsieme $S \subset X$ denso in $X$ e numerabile." Based on Google Translate is seems that this is definition of separable space and also an alternative name third axiom of countability is used for such spaces.