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One way to define a topology for a set (of "points") X is to first give a suitable definition of a neighborhood of $x$, for any point $x \in X$, and then to define an open set as any subset of $X$ that is a neighborhood of all of its elements. A topology for $X$ is then defined as the family of all open subsets of $X$.

I would like to know who first proposed this particular approach to defining a topology.

(Note that another approach first defines open sets, and then defines neighborhoods in term of open sets. Yet another approach takes the closure of a set as the fundamental definition. Etc.)

EDIT: The motivation behind this question can be found in this other question. In that question I give a verbatim rendition of a set of axioms defining neighborhoods and then defining open sets in terms of neighborhoods. I argue that these axioms, as stated, are faulty. Later (in my answer to the question) I show what I believe is the genesis of the error. The problem lies in the subtle difference between the following two alternative versions of one of the axioms:

(B$3$) for each $x$, the set of neighborhoods of $x$ is closed under non-infinite intersections;

(B$3^{\prime}$) if $U_1, \dots, U_n$ are neighborhoods of $x$, then $U_1 \cap \dots \cap U_n$ is a neighborhood of $x$;

(Of course, one typically writes finite instead of the awkward non-infinite, but I've used the latter in (B$3$) to bring out the fact that the axiom applies to the empty intersection as well. IOW, the axiom implies that, for each $x \in X$, the entire set $X$ is a neighborhood of $x$.)

In fact I've come across even more emphatic versions of (B$3^\prime$):

(B$3^{\prime\prime}$) if $U_1$ and $U_2$ are neighborhoods of $x$, then $U_1 \cap U_2$ is a neighborhood of $x$;

The problem with (B$3^\prime$) and (B$3^{\prime\prime}$) is that they fail to rule out the case in which some $x \in X$ has no neighborhoods at all, which in turn will imply that the entire space $X$ is not open.

My interest in this question is that I have found at least three separate mentions in the literature of the faulty set of axioms, one as recent as 1997. It appears that this is a case of an error that has been propagated for decades, and I'm curious about its origin. In fact, it's not hard to imagine a sequence of rewordings $\text{B}3 \to \text{B}3^\prime \to \text{B}3^{\prime\prime}$.

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I suspect that it was the other way around. Neighborhoods were used to define continuity ($\epsilon-\delta$ is essentially about neighborhoods) and somebody saw that you could generalize this to something they called a "topology." So what you really are looking for is who first thought of the "closed/open" definition of "topology." – Thomas Andrews May 28 '13 at 13:30
@ThomasAndrews: I'm not asking what was the first approach to defining a topology, but rather who was the one who first proposed the idea of taking neighborhoods as the fundamental definition. – kjo May 28 '13 at 13:32
When Felix Hausdorff defined topological spaces 1914 in his book "Grundzüge der Mengenlehre" (he exactly defined what is now called $T_2$- or Hausdorff spaces), we used neighbourhoods. So as far as I now, the first definition of an (almost) general topolgical space used nhoods and was by Hausdorff. – martini May 28 '13 at 13:37
@roger: see – kjo May 28 '13 at 13:38
@roger: it's what you do when you consider the notion of base and the topology generated by a base. – Martin Argerami May 28 '13 at 13:40

Felix Hausdorff wrote 1914 in his book "Grundzüge der Mengenlehre":

(A) Jedem Punkt entspricht mindestens eine Umgebung $U_x$; jede Umgebung $U_x$ enthält den Punkt $x$.
(B) Sind $U_x$, $V_x$ zwei Umgebungen desselben Punktes $x$, so gibt es eine Umgebung $W_x$, die Teilmenge von beiden ist ($W_x \subseteqq \mathfrak D(U_x, V_x)$).
(C) Liegt der Punkt $y$ in $U_x$, so gibt es eine Umgebung $U_y$, die Teilmenge von $U_x$ ist ($U_y \subseteqq U_x$).
(D) Für zwei verschiedene Punkte $x,y$ gibt es zwei Umgebungen $U_x, U_y$ ohne gemeinsamen Punkt ($\mathfrak D(U_x, U_y) = 0$).

My translation:

(A) Each point has at least one neighbourhood $U_x$; each neighbourhood $U_x$ contains $x$.
(B) If $U_x$, $V_x$ are two neighbourhoods of the same point $x$, then there is a neighbourhood $W_x$, which is a subset of both ($W_x \subseteqq \mathfrak D(U_x, V_x)$).
(C) If $y$ is a point of $U_x$, then there is a neighbourhood $U_y$, which is a subset of $U_x$ ($U_y \subseteqq U_x$).
(D) Each two different points $x,y$ have neighbourhoods $U_x$, $U_y$ which do not have a common point ($\mathfrak D(U_x, U_y) = 0$).

As far as I know this is one the earliest definitions of the concept "topological space" (to be exact, of a Hausdorff space due to axiom (D)) and the first which uses neighbourhoods (note: $\mathfrak D$ denotes the intersection herein, it is the first letter of the german "Durchschnitt").

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Note that, because of (C), we have a system of open neighbourhoods that is defined here, so a (base of) open neighbourhoods is the notion that is used as axiomatic. Not the later notion of system of neighbourhoods (as sets that contain an open set around that point). – Henno Brandsma May 28 '13 at 17:41

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