Definition Topological Space Consider the definition of a topological space:
Topological Space: A topology on a set $X$ is a collection $\mathcal{T}$ of subsets of $X$ such that 


*

*$\emptyset,X \in \mathcal{T}$.

*The union of elements of any subcollection of $\mathcal{T}$ is in $\mathcal{T}$.

*The intersection of the elements of any finite subcollection of $\mathcal{T}$ is in $\mathcal{T}$.
Then a topological space is the ordered pair $(X,\mathcal{T})$ consisting of a set $X$ and a topology $\mathcal{T}$ on $X$. 
When I introduce this idea to students, it is simple enough to motivate why this definition was selected using elementary Calculus concepts and the fact that Topology originally was supposed to generalize these ideas. It makes this definition more accessible to them. Of course, one can equally put the definition in terms of closed sets. 
However, I have been asked why this definition. Meaning, not why this definition
as compared to the closed set definition but rather why this definition works better than other possible definitions. Then the student asks what other definitions were tried and why did they fail so that the above definition was settled on. 
I've thought about this myself but I've never found a reference that talks about how this definition was the the one a mathematician would want compared to other possible definitions. I'd love to be able to have a good answer for students for this and to be able to include this in why the above definition is 'good'. Does anyone know of a historical explanation in any text or paper that discusses, if only briefly, other definitions that were originally tried? I have not been successful in finding such sources. Even thoughts on possible alternate definitions of a topology that turn out to be 'inferior' to the standard one with a brief explanation why would be acceptable in lieu of a source.
 A: Several people were involved in the early development of general point-set topology, but a large part of it is due to Felix Hausdorff, and can be found in his Grundzüge der Mengenlehre from 1914. (Note that the English edition is translated from a later edition which was much rewritten.)
Hausdorff noted that there are three basic concepts by means of which one can base a general theory of topological spaces: the notion of distance, neighborhoods, and limits. Starting with a notion of distance, one can derive the other two (as Fréchet had done in 1906), and starting from a notion of neighborhood, one can define limits. Hausdorff chose to define topological spaces in terms of neighborhood axioms, together with what we now call the Hausdorff separation axiom. 
In the years following Hausdorff's book, different variations in the definition of a topological space were explored. Kuratowski gave a definition in terms of closure axioms. Tietze gave a definition in terms of open sets. Several text books were published in the 1930's with slightly different choices of separation axioms, etc. Finally, in 1940, Bourbaki (or André Weil and Claude Chevalley) chose our modern definition, which was also used in the influential book by John Kelley (1955).
A: All that I write is my opinion but not anything in some sources you described.
We recall the properties of open set of the real number set $\mathbb{R}$:


*

*$\Phi$, $\mathbb{R}$ are open sets.

*The intersection of the elements of any finite open sets is open set.

*The union of elements of any open sets is open set.
The definition of topology space is developed from these properties. All the fundamental definitions (neighborhood, convergence, continue, cover...) in $\mathbb{R}$ can be described by open set but don't need the definition of "distance". For talking about the definitions and properties of any space $X$, we make an abstract definition from the open set of $\mathbb{R}$ and get the definition of topology space. In fact, this is a development of mathematics. The definitions of topology is not a figment of imagination. And I didn't hear some other definitions.
