Let $X$ and $Y$ two sets. Given a set $\mathcal{D}\in\mathcal{P}(\mathcal{P}(X))$ one can define a relation $R$ on $Y^X$ saying $fRg$ iff $\{x\in X|f(x)=g(x)\}\in\mathcal{D}$. It is not too hard to show that if $card(Y)\geq3$, $R$ is an equivalence relation iff $\mathcal{D}$ is a filter. Let us call $\tilde{f}$ the equivalence class of $f$. Philosophically speaking, the sets in $\mathcal{D}$ are sets which are more important than others and $\tilde{f}=\tilde{g}$ iff they are equal on some important subset of $X$.

Now, recall the usual definition of a limit following a filter $\mathcal{F}$ on $X$, if $l\in Y$ with $Y$ a topological space : $$f\xrightarrow[\mathcal{F}]{} l \Leftrightarrow \forall V\in\mathcal{V}_l,\exists F\in\mathcal{F},f(F)\subset V$$ What we see here is that a limit is primarily concerned not with $f$, but with its class $\tilde{f}$, as $(fRg \wedge f\xrightarrow[\mathcal{F}]{} l) \Rightarrow g\xrightarrow[\mathcal{F}]{} l$. It makes then sense that we use filters to see the spot where a limit is taken. When we use the Fréchet filter on $\mathbb{N}$, the values of the sequence for $n\leq$ a given $n_0$ do not matter. The same goes in a metric space when $x\rightarrow x_0$, the value of the function outside of a given $V\in \mathcal{V}_x$ do not matter.

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. $\forall V\in\mathcal{V}_x,\exists W\in \mathcal{V}_x, W\subset V\wedge \forall y\in W,W\in \mathcal{V}_y$

I will try to give an explanation of the first two axioms in light of the discussion of filters given above.

1. The first axiom lets one see that we can use the neighbourhood of $x$ as the spot where the limit is taken.
2. One can prove the following for a set $X$ and a filter $\mathcal{F}$ : $\mathcal{F}\subset F_x$, the principal ultrafilter for some $x$ or the Frechet filter $\mathcal{F}_{Frechet}\subset \mathcal{F}$. It makes sense that we use the former possibility when we want to define nearness to $x$. Furthermore, if the second axiom was false, there would be an important set which does not contain $x$.
3. 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.