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.