# Topologies without the axiom that finite intersection of open sets is open

A topology is a pair of

• a nonempty set $$P$$ of points, and
• a set $$Opens\subseteq 2^P$$ of open sets that is closed under two closure conditions:
1. arbitrary (possibly infinite) unions and
2. finite (possibly empty) intersections.

So far, so standard. Please now consider dropping closure condition 2 above (closure under finite intersections).

My question is: can anyone provide a reference or references for any early discussion of the notion of topological space without condition 2 above. It was eventually included in the standard definition as we know, but does anyone know of references to any debate over whether to include it or not (much as there were discussions over, for example, the axioms of set theory).

Thanks in advance.

• Off-hand I don't know the answer to your question other than your focus is a little ahistorical. The formulation we now use as conditions satisfied by a collection of so-called open sets was settled on several decades after many other approaches had been worked with by researchers -- some equivalent to our present notion, some nonequivalent that imply our present notion, probably some nonequivalent that are implied by our present notion, and some that neither imply nor are implied by our present notion. Sep 14, 2022 at 13:34
• For an example of how your focus is ahistorical, see the first few pages of this 1929 paper by Chittenden, which is somewhat significant in the early history of (general or point set) topology. For some various ways in which the present day notion of a topology can be defined, see this MSE question, and if you want to drill down on one of these ways (the closure operator), see the 3-part answer to this mathoverflow question. Sep 14, 2022 at 13:47
• I know you can have a base for a topology where all the base's elements are mutually disjoint, obviating the need for discussing intersections, but that still meets condition 2. Sep 14, 2022 at 23:30
• Why would you call this a topology? It's just then a join complete distributive lattice. A much more productive notion is to drop the notion of points. This gives the notion of 'pointless' topology. This is appealing metaphysically where one doubts the existence of points. Sep 23, 2022 at 9:39

## 1 Answer

There was no early discussion of topology without closure under finite intersections. The open set definition first given by Bourbaki (1940) includes this condition, and its predecessors in terms of neighborhoods (Hausdorff, 1914) and in terms of closures (Alexandrov-Hopf, 1935) include its equivalents, see History of various definitions of topology and references there.

There was long experimentation (from 1906 to 1940) with various approaches to defining topology (via accumulation points, neighborhoods, closures, open sets), and discussions of whether to include various separation axioms (by Hausdorff in 1927 and Alexandrov-Hopf in 1935) or some notion of uniformity (by Weil in 1937), but dropping closure under finite intersections (or its equivalents) was not considered. The remaining condition would have been too weak for intended applications.