What does the 3rd axiom of topologies defined by neighbourhood mean? [closed]

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.

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closed as unclear what you're asking by Andres Caicedo, Ryan Budney, Neil Strickland, S. Carnahan♦May 5 '14 at 18:13

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You seem to be rediscovering Bourbaki's Proposition 2 here... –  Francois Ziegler May 4 '14 at 23:20
My question is more on terms of intuition and meaning (here limits and continuity are the most important facts as far as a topology is concerned) than formal equivalence of definitions. So the fact that there is such an equivalence does not matter in that context. This is why I asked the question. –  Florian May 4 '14 at 23:24
What do you mean "need"? What is your goal and what do you think we already "need"? –  Monroe Eskew May 4 '14 at 23:58
Possible duplicate: mathoverflow.net/questions/19152/… –  David White May 5 '14 at 1:38
The third axiom says that any neighbourhood of a point $x$ is also a neighbourhood of all the points "sufficiently near" to $x$; the reason to introduce it is that axioms 1 and 2 do not put any relation among filters at different points, hence precluding most of the local-global machinery so fruitful in topology. –  johndoe May 6 '14 at 6:14

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. 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 and pseudotopological spaces, 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; 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 emphasized by Ronnie Brown, Norman Steenrod, and others).

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Of course, this answer doesn't explain yet what's up with that third axiom. I'd be inclined to compare it to the characterization of topological spaces among pseudotopological spaces along the lines of ncatlab.org/nlab/show/relational+beta-module. To be brief: axiom 2 of the OP is compared to the condition that the principal ultrafilter at a point $x$ converges to $x$. Regarding ultrafilter convergence as a morphism $\xi: \beta X \to X$ in the (bi)category of relations $Rel$, this becomes a unit condition on $\xi$ relative to a ultrafilter monad structure $\beta$ on $Rel$. (Cont.) –  Todd Trimble May 5 '14 at 10:48
(Cont.) Thus pseudotopological spaces are sets $X$ equipped with a relation $\xi: \beta X \to X$ satisfying a unit condition. What precisely carves out topological spaces among pseudotopological spaces is the imposition of an extra associativity or transitivity condition on $\xi$, a kind of lax version of the associativity condition for algebras over a monad, as explained in the relational beta modules article at the nLab. I submit that the third axiom of the OP probably bears comparison with that transitivity condition, but it would take some time to explain that carefully. –  Todd Trimble May 5 '14 at 10:53
I am sorry, but I would really want to build an intuition on topological spaces (which, I admit, was not clear from the first formulation of my question). Your argument is very interesting and shows a real erudition, but it seems rather involved. –  Florian May 5 '14 at 20:19
Well, admittedly I hadn't tried hard to condense it down into something more reader-friendly; I was jotting down some observations with the hope of maybe coming back to it later. Although this relational beta-module viewpoint on topological spaces is not a bad thing to contemplate, when you find the time and interest... –  Todd Trimble May 5 '14 at 20:28