Recall the axioms of a topology defined in terms of neighbourhoods, we call a topology on $X$ a family $(\mathcal{V}\_x)\_{x\in X}$ of sets in $\mathcal{P}(\mathcal{P}(X))$ which verifies for all $x\in X$ :


1. $\mathcal{V}\_x$ is a filter on $X$
2. $\forall V\in\mathcal{V}\_x,x\in V$
3. $\forall V\in\mathcal{V}\_x,\exists W\in \mathcal{V}\_x, W\subset V\wedge \forall y\in W,W\in \mathcal{V}\_y$


**What meaning do you give to the third axiom ?** I see that it guarantees the equivalence between the usual axioms of a topology using open sets and the ones presented above. But I want more than a mere formal equivalence of definitions. I want something which has real meaning as far as limits are concerned, in order to build an intuition of topological spaces (which I think the above discussion begins to give). I want to have what I have for many other structures : a vision.



What does the 3rd axiom of topologies defined by neighbourhood mean?
Since it is the third axiom that is crucial in establishing the connection with the traditional notion of topology, one way of understanding the last question (on the "need for" open sets) is by asking a related question: what would happen if we simply dropped the third axiom? We wouldn't get the notion of topological space; we'd get some other more general notion of "space". Would there be a point to such a notion? 


It seems to me the resulting notion is precisely that of [pretopological space](http://ncatlab.org/nlab/show/pretopological+space). This is one of several generalizations in which one can "do topology", i.e., the classical notions of Hausdorffness, compactness, etc. can be formulated in the context of pretopological spaces. The same can be said for increasingly more general notions like [convergence spaces](http://ncatlab.org/nlab/show/convergence+space) and [pseudotopological spaces](http://ncatlab.org/nlab/show/pseudotopological+space), which are based on notions of filter convergence. 


As for what would be the point of such generalization (except generalization for its own sake): one of the chief annoyances of the category $Top$ of classical topological spaces is the lack of a useful general notion of function space. I an not sure about the case of pretopological spaces, but what it remarkable about the categories of convergence spaces and pseudotopological spaces is that they form [quasitoposes](http://ncatlab.org/nlab/show/quasitopos); here the key property is that not only are they cartesian closed (thus having good function spaces), but so are all their slice categories (where we look at categories of such spaces over a suitable base space). This makes them convenient for many purposes (recalling the sense of [convenience](http://ncatlab.org/nlab/show/convenient+category+of+topological+spaces) emphasized by Ronnie Brown, Norman Steenrod, and others). 


