Foundations of topology I recently went to a talk of Oleg Viro where he expressed his dissatisfaction with current foundations of differential topology parallel to what has been discussed here. 
Also some time ago I read about Grothendieck's "Denunciation of so-called “general” topology" with interesting comments also made here:

According to Winfried Scharlau's book, Grothendieck described his work
  in a letter to Jun-Ichi Yamashita as: "some altogether different
  foundations of 'topology', starting with the 'geometrical objects' or
  'figures', rather than starting with a set of 'points' and some kind
  of notion of 'limit' or equivalently) 'neighbourhoods'. Like the
  language of topoi (and unlike 'tame topology'), it is a kind of
  topology 'without points' - a direct approach to 'shape'. ...
  appropriate for dealing with finite spaces...

So I am wondering what progress has been made here and in what directions. Does there currently exist an approach to the foundations of general topology that is not based on a notion of "points", in the spirit of Grothendieck's denunciation?
 A: There are different ways to respond to Grothendieck's challenge, depending on how exactly you interpret what he is looking for. As mentioned by ACL in a comment, the theory of o-minimality provides one way to formalize the concept of "tameness" that Grothendieck sought. This approach is discussed in some detail by Norbert A'Campo, Lizhen Ji, and Athanase Papadopoulos in their paper, On Grothendieck's tame topology.  See also Brian Tyrell's 2017 thesis, An analysis of tame topology using o-minimality.
But another take on the question is to focus on Grothendieck's remark that he favors an axiomatic approach, and that "once this necessary foundational work has been completed, there will appear not one “tame theory”,
but a vast infinity." This remark brings to mind something called synthetic mathematics, in which mathematical objects are not built up from a set-theoretic "substrate" but are studied axiomatically.  Examples include homotopy type theory and synthetic differential geometry, which can be developed without ever building a "model," and which in fact admit many different "models."
A related idea is that topology traditionally combines two ostensibly distinct concepts: cohesion (or neighborliness) and shape, and that we might be better off developing these concepts separately rather than combining them. I listened to a talk by Emily Riehl (Elements of ∞-Category Theory, 17 Feb 2021), in which she made some interesting comments along these lines (here and here).  The Brouwer fixed-point theorem is is vacuous if you regard homotopically equivalent spaces as the same; by contrast, the computation of the higher homotopy groups of spheres is arguably a question about homotopy types and not about point sets. You could interpret Grothendieck as calling for a separation between these two "kinds of topology," because if you commingle them then sometimes the definitions from point-set topology can introduce annoying technicalities that are irrelevant to the questions Grothendieck was primarily interested in.  According to Riehl, condensed mathematics and the closely related theory of pyknotic sets develop this idea further.  Fittingly, they build on the concept of a Grothendieck topology but take it further.
A: Reading section 5 in Grothendieck's essay Esquisse d'un programme it becomes clear that with regard to topology Grothendieck was bothered by some artificial foundational problems introduced by the fact that the foundations of topology were created by analysts rather than by geometers and topologists. Specifically he refers to phenomena such as space-filling curves which he thinks should be ruled out at the foundational level by a more careful choice of definitions of the basic objects we work with.
The basic model is Hironaka's semianalytic sets (or what Grothendieck proposes to call piecewise analytic sets) where such phenomena do not occur, and which on the other hand is sufficiently rich to accomodate various constructions in geometry and topology, such as coning, stratification, etc.  What Grothendieck seeks to do is provide an axiomatisation that would be more or less satisfied by Hironaka's proposal, but that would be realizable in other models as well.  Notes Grothendieck:
This situation, like so often already in the history of our science,
simply reveals the almost insurmountable inertia of the mind, burdened by
a heavy weight of conditioning, which makes it difficult to take a real look
at a foundational question, thus at the context in which we live, breathe,
work – accepting it, rather, as immutable data.
My conclusion is that Grothendieck's proposal in this context does not necessarily amount to a search for a foundation not based on points.  Rather the idea is to get away from the continuous category with its odd phenomena that are viewed by Grothendieck as being a function of inadequate foundations rather than intrinsic mathematical merit. Not an uncommon phenomenon I must say.
