History of various definitions of topology

Question

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.

Answer 1

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].