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*}
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.
Best Answer
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.