TITLE: Why the axioms for a topological space are those axioms?
QUESTION [13 upvotes]: This question might have even been asked here before, I don't really know, so sorry if it's duplicate. I've started to study topological spaces and I've found the axioms for such spaces kind of hard to motivate. Well, the ideia of a metric space is much easier to motivate: "the concept of distance is context dependent, so we want a general idea of what distance is and a general idea of a set on which we can measure distances". The axioms then for a metric are very intuitive, easy to motivate and everything else.
Then we start studying properties of subsets of metric spaces. We define open balls as a way to make precise the notion of "the set of all points that are sufficiently close to a central point" and we define open sets to make precise  "sets such that for every point, other points sufficiently close are also in the set", which can also be thought as sets such that each point can be oscilated a little bit from it's position and the point will stay on the set.
After that we can define lots of things: limit points, closed sets, dense sets, perfect sets, compactness and so on. We also see that all of those notions can be made precise mentioning the open sets alone: the metric is not really necessary to talk about those things, as soon as we can define what open sets are. So this is enough motivation to define a structure on which we have open sets.
The answer to this problem is to define a topology on a set $X$ as a set $\mathcal{T} \subset \mathcal{P}(X)$ such that $\mathcal{T}$ is closed under arbitrary unions, finite intersections and such that $X,\emptyset \in \mathcal{T}$. If, $X$ is a metric space, and we let $\mathcal{T}$ be the set of open sets as they are defined using balls, then the three properties are satisfied.
My only question is: why those properties capture completely the idea of an open set? I mean, amongst all properties of open sets, why do we select those three? I've always heard that topology is meant to study qualitatively global properties of forms, that's the way we start in $\mathbb{R}^n$ and the way we generalize to metric spaces: we introduce tools that allows us to define carefully some of these properties and we work out definitions. It doesn't seem clear the connection of this motivation for topology and the actual definition.
I've seem a similar question on MathOverflow, and there was one answer trying to motivate this in terms of rulers, but I really didn't get the idea. Can someone give a little help with this?
Thanks very much in advance!

REPLY [5 votes]: I agree with Andreas. 
The open set axioms for a topology are usually presented as the axioms for a topology, but in fact there are as number of equivalent ways of defining a topology, via neighbourhoods, open sets, closed sets, closure, and even interior or boundary. The neighbourhood axioms are surely the most intuitive to a beginner, and most clearly related to the usual notion of continuity in analysis. Indeed the definition: "$f$ is continuous at a point $x$ of its domain if for all neighbourhoods $N$ of $f(x)$ there is a neighbourhood $M$ of $x$ such that $f(M) \subseteq N$" is more direct and geometric than the $\epsilon$-$\delta$ definition, since the latter are just measures of the sizes of  neighbourhoods, and so a step away from the geometric picture of a neighbourhood. Those numbers  $\epsilon, \delta$ are of course a useful computational representation, as numbers are. 
Of course the open set axioms are very elegant and logically simpler than the neighbourhood axioms, and are in some case the best way of defining a topology, for example for identification spaces. But beginners should be encouraged to make their own judgement on the advantages and disadvantages of each definition of a topology. 
Part of the need for such abstraction from the notion of metric space was the development of ideas of Riemann surfaces, and then the  notion of manifold. The work of Hausdorff was a key in this development. 
January, 2017  I recommend here  the Introduction to Peter Freyd's Abelian Categories part of which states: "If topology were defined as the  study of families of sets closed under finite intersection and infinite unions a serious disservice would be perpetrated  on embryonic students of topology......A better (albeit not perfect) description of topology is that it is the study of continuous maps;..."
This reminds me of a quotation of Einstein: "Concepts which have proved useful for ordering things easily assume so great an authority over us, that we forget their terrestrial origin and accept them as unalterable facts.  $\ldots$  It is therefore not just an idle game to exercise our ability to analyse familiar concepts, and to demonstrate the conditions on which their justification and usefulness depend, and the way in which these developed, little by little $\ldots$" Difficulties in algebraic topology with the notion of "bare" topological space, i.e. with no extra structure, are discussed in this paper.