Some questions about different axiomatic systems for neighbourhoods

Hausdorff axiomatises a set of "basic open neighbourhoods" of $x$ essentially, while the other one axiomatises the more general notion of neighbourhood ($N$ is a neighbourhood of $x$ iff there is an open subset $O$ with $x \in O \subseteq N$), which form a non-empty filter at each point (which is the summary of axioms (0)-(3) ) and (4) is needed to couple the different neighbourhood systems and make a link to openness: it essentially says that every neighbourhood contains an open neighbourhood (one that is a neighbourhood of each of its points), it ensures that when we define the topology in the usual way from the neighbourhood systems, that $\mathcal{N}_x$ becomes exactly the set of neighbourhoods of $x$ in the newly defined topology too. I gave that proof in full on this site before. See this shorter one and this longer one, e.g.

The axiom systems are not equivalent. By (C), every neighborhood $U_x$ is an open set. On the other hand, (3) implies that $\mathcal{N}(x)$ is the family of all neighborhoods of $x$, and (4) assures that the members of $\mathcal{N}(x)$ are indeed neighborhoods, i.e. contains $x$ in the interior. So the first system are axioms for (Hausdorff) open neighborhood base, while the second system are axioms for complete neighborhood system. There are also axioms for neighborhood base system, which are slightly weaker than both of these.

There are various similar axiom systems. In general, you have families of sets $\mathcal{N}(x)$ for $x ∈ X$, and you want to induce a topology as follows: $U ⊆ X$ is open if and only if for every $x ∈ U$ there is $N ∈ \mathcal{N}(x)$ such that $N ⊆ U$. There is a weak set of axioms that assures that this indeed induces a topology. But you may add more axioms if you want more properties like

• each $N ∈ \mathcal{N}(x)$ is a neighborhood of $x$ (this is not automatical);
• each $N ∈ \mathcal{N}(x)$ is open;
• each neighborhood of $x$ is a memnber of $N ∈ \mathcal{N}(x)$.