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  • $\begingroup$ I use Russian translation that is a merged version (by P. Alexandrov and A. Kolmogorov) of Hausdorff 1914 and 1927 books. I tried to find the corresponding pages in the original without success (archive.org/details/grundzgedermen00hausuoft/page/212). But I cannot read Deutsch. So, probably, the example was introduced by P. Alexandroff or A. Kolmogorov (I can send a scan if it helps). Still, how such `general' t.s. are called these days? $\endgroup$ Commented Sep 22, 2019 at 22:50
  • $\begingroup$ @BjørnKjos-Hanssen, thank you - I've refined the question. $\endgroup$ Commented Sep 22, 2019 at 23:23
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    $\begingroup$ @ToddTrimble, you've got it right. "But I don't see how the complementation operator deserves to be called topological" - me either, but I leaving room for doubt and trying to clear it up. $\endgroup$ Commented Sep 24, 2019 at 3:45
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    $\begingroup$ I don't know about Hausdorff's book. I'm fairly familiar with the 3rd edition, because an English translation exists (a copy of which I've had for several decades), but not the 1st edition, which includes a lot of topics omitted in the 2nd and 3rd editions. However, Introduction to General Topology by Zlatko P Mamuzić (1963; review on pp. 127-128 here) begins by considering "closure operators" satisfying NO axioms (continued) $\endgroup$ Commented Sep 24, 2019 at 8:12
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    $\begingroup$ (the complement of a set is the 3rd example on p. 13, lines 7-9 from bottom) and then adds various axioms (e.g., empty set maps to empty set, each set is a subset of its "closure", the "closure" of the union of two sets is a subset of the union of the closures of the two sets, etc.) and discusses the relationships among these axioms. A better known text, Topological Spaces by Eduard Čech (1966), is a very complete study spaces in which idempotency is not assumed for the closure operation. See also this. $\endgroup$ Commented Sep 24, 2019 at 8:24