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.

  • $\begingroup$ 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

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].

  • 3
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.