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This question is not actually a serious mathematical one. I have been reading Point Set Topology for a while, and turns out that there are various possible ways to define a topology. Most popular one is using open set axioms. Another one is using closure axioms, which was introduced by Kuratowski.

I am interested to know name of the researchers who had given the definition of topological spaces in the following ways:

  1. Definition through Open sets (The most popular one)
  2. Definition through neighbourhood system
  3. Definition through interior points

Thanks in advance.

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  • $\begingroup$ Maybe HSM.SE is a more appropriate venue for this question? $\endgroup$ – YuiTo Cheng Mar 23 at 10:55
  • $\begingroup$ There is a book on these kind of questions : the Handbook of the History of General Topology", which answers all those questions (publisher link). Go find it in a good library. $\endgroup$ – Henno Brandsma Mar 24 at 6:25
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From "The emergence of open sets, closed sets, and limit points in analysis and topology" [PDF], by Gregory H. Moore:

The idea of an open set in an abstract space (as opposed to n-dimensional Euclidean space, where the idea was due to Baire and Lebesgue) was originated by Felix Hausdorff in the context of his topological spaces. However, what Hausdorff called a topological space is a more specialized idea than what is now universally called a topological space. What he used as a primitive idea is “neighborhood of a point.” ... Immediately after giving his axioms for a topological space, Hausdorff defined what he meant by an “interior point” of a subset A of a topological space. Namely, x is an interior point of A if some neighborhood of x is a subset of A. And x was said to be a boundary point of A if x belongs to A but is not an interior point of A. Then a set A was defined to be an open set (“Gebiet”) if all of its points are interior points [1914, 214–215].

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    $\begingroup$ In the monumental General Topology by R. Engelking, at the end of many sections there is a "Historical Notes And Bibliography" citing original sources, which are assembled in a thorough index of references. $\endgroup$ – DanielWainfleet Mar 23 at 15:28

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