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In What is elementary geometry? (pdf) Alfred Tarski defined elementary geometry to be

that part of Euclidean geometry which can be formulated and established without the help of any set-theoretical devices.

I wonder whether there is any fragment of topology that deserves the name elementary in the aforementioned sense. Of course, the notion of set is crucial for development of modern topology (with the whole retinue of various kinds of sets defined via topological notions), but maybe there are some branches of mathematics that would deserve the name.

Specifically I would like to ask the following:

Is the theory of frames and locales (in which the countepart of the notion of open set is assumed as basic) a reasonable candidate for elementary topology in the aforementioned sense? Or which part(s) of this theory could be treated as candidate(s)?

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    $\begingroup$ Are you aware of pointless topology? $\endgroup$
    – Nate Eldredge
    Feb 25, 2016 at 18:10
  • $\begingroup$ Yes, I am. I will add it to candidates. But I think that only this part which is done without the help of category theory is a candidate for elementary topology. $\endgroup$
    – Rafał Gruszczyński
    Feb 25, 2016 at 18:11
  • $\begingroup$ So you want an axiom system specific for basic topology, that is not readily transcribed into some other standard axiom system for all of mathematics? I imagine you could so such for pretty much any subject in mathematics. You just have to pick the objects you consider important and spend the time building an axiom system. Or perhaps you are interested only in what has been done, at present? $\endgroup$
    – Ryan Budney
    Feb 25, 2016 at 20:18
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    $\begingroup$ Actually, considering the fact that elementary geometry is about a first-order theory of Euclidean geometry (which distinguishes it from Hilbert's axioms, and ties geometry in closely with Tarski's development of real closed fields, quantifier elimination, etc.), I think maybe closer to what you might be interested in is tame topology. See the wonderful little book Tame Topology and O-minimal Structures by Lou van den Dries, which develops definably continuous functions, definable stratifications, and so on. $\endgroup$
    – Todd Trimble ♦
    Feb 26, 2016 at 15:25
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    $\begingroup$ The trouble with frames is that they cannot be defined in a truly 'elementary' way: one wants arbitrary sups, which involves quantifying over subsets and so one is no longer dealing with first-order theories. One can extract some first-order aspects, such as modal operators on Heyting algebras, but I think that would be a somewhat poor approximation to what we mean by 'topology'. $\endgroup$
    – Todd Trimble ♦
    Feb 26, 2016 at 15:36

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