Some questions about different axiomatic systems for neighbourhoods

Question

I was thinking a few days ago about the development of topology, especially how you arrive at the concept of a topology $\tau$. I knew that a lot of initial ideas came from Hausdorff who defined a topological space by giving neighbourhood axioms; so I had a look at the original text ,,Grundzüge der Mengenlehre'' to see how his definition compares to the contemporary one.

Here is a translation of the ,,Umgebungsaxiome'' that Hausdorff gives:

$(A)~$ To every point $x$, there is some neighbourhood $U_x$; every neighbourhood $U_x$ contains the point $x$.

$(B)~$ If $U_x,V_x$ are two neighbourhoods of the same point $x$, there is a neighbourhood $W_x$, which is in both of them ($W_x \subseteq U_x \cap V_x$).

$(C)~$ If the point $y$ lies in $U_x$, there is a neighbourhood $U_y$, which is a subset of $U_x$ ($U_y \subseteq U_x$).

$(D)~$ For two different points $x,y$ there exist two neighbourhoods $U_x, U_y$ with no common points ($U_x \cap U_y = \emptyset$).

and here is a version of the neighbourhood axioms you might find in a modern textbook

$\mathcal{N}(x)$ is a set of neighbourhoods for $x$ iff \begin{align*} (0)&~~~ x \in \bigcap \mathcal{N}(x) \\ (1)&~~~ X \in \mathcal{N}(x) \\ (2)&~~~ \forall ~U_1,U_2 \in \mathcal{N}(x) : ~ U_1 \cap U_2 \in \mathcal{N}(x) \\ (3)&~~~ \forall~ U \subseteq X ~~\forall~ N \in \mathcal{N}(x):~ N \subseteq U \Longrightarrow U \in \mathcal{N}(x) \\ (4)&~~~ \forall~ U \in \mathcal{N}(x) ~~\exists~ V \in \mathcal{N}(x)~ \forall p \in V :~ U \in \mathcal{N}(p) \end{align*}

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Here are a few questions I still have after reading and thinking about it:

$(i)$ Are these axiomatic systems equivalent? Even leaving out axiom $(D)$ (since it says that the space is $T_2$) it does not seem to me that they are. A few of them clearly are equivalent, but I don't see how you could derive $(3)$ from $(A) - (C)$. I could at least imagine that one was added over time, which would explain the problem.

$(ii)$ What is the use of axiom $(4)$? I think most helpful to me would be an example of a proof in which the axiom is indispensable.

$(iii)$ (more historically minded question) How did the word ,,Umgebung'' ended up being translated to neighbourhood?

Hausdorff calls a set $U_x$ a ,,Umgebung'' of $x$ which in English (at least in the mathematical literature) is called a neighbourhood. This is quite strange, since neighbourhood has a direct translation to the German: ,,Nachbarschaft'' whose meaning is quite different from the one of ,,Umgebung''. I my opinion the later would be better translated by the word surrounding. Which makes me curious how the translation came about.

Answer 1

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.

Answer 2

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 $N ∈ \mathcal{N}(x)$.