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.

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    $\begingroup$ I’ve never seen it (except in that question). $\endgroup$ – Brian M. Scott Jun 3 '15 at 16:18
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    $\begingroup$ @BrianM.Scott Ditto here. And I wonder if there are more than $N_1$ for presumably first countable and this $N_2$ for second countable. I'd rather use $w(X) = \aleph_0$ and $\chi(X) = \aleph_0$ and have other "countabiliy axioms" be defined by the countable values of other cardinal invariants (pseudo character, tightness, network weight etc.) What would $N_3$ then be, or $N_0$? A hierarchy of 2 isn't much of an hierarchy. $\endgroup$ – Henno Brandsma Jun 3 '15 at 19:51
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    $\begingroup$ This question also uses $N_1$ forst first countable. $\endgroup$ – Martin Sleziak Dec 31 '16 at 7:56
  • $\begingroup$ The 1975 survey paper by Frank Siwiec "Generalizations of the First Axiom of Countability" has an extensive "dictionary" of defined terminology in Section 1, intended as a convenient reference for the later sections. Nowhere is a "hierarchy" of such generalizations mentioned in this paper. $\endgroup$ – hardmath Nov 28 '17 at 16:15
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    $\begingroup$ This seems to be a matter of "local custom". I get a few relevant hits with "axioma de numerabilidad" N1 (spanish), but more with "Abzählbarkeitsaxiom" A1 (german). The latter gives me some textbooks too. $\endgroup$ – Niels J. Diepeveen Nov 29 '17 at 18:28

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