Let me point out that this appendix in MacPherson's notes is explicitly mentioned in a wonderful and highly accessible book by the model theorist Lou van den Dries, Tame Topology and O-minimal Structures (page 8). Indeed, the entire corpus of o-minimal geometry can be viewed as giving a precise response to the frequently expressed desire, perhaps most eloquently enunciated in Grothendieck's *Esquisse d'un Programme*, to put the sort of "tame topology" that MacPherson is pointing to on firm theoretical ground.

Where MacPherson says that only finitely many data are required to define a finite topological type (FTT), he says he means subsets of manifolds -- probably we can assume the manifolds are Euclidean spaces $\mathbb{R}^n$ without any real loss of generality -- and a reasonable guess is that he means the data are specified by finitely many conditions, for example a subset carved out by finitely many equalities and inequalities involving some basic staple functions like polynomials should qualify as an FTT. Which functions can be admitted is presumably open to discussion, so long as finite expressions involving them do not lead to things like the Cantor set being "finitely definable", which for the purposes of this discussion will be considered "pathological".

There are a number of formalisms which capture this intuition in one way or another; the best known or most investigated is probably that of o-minimal structures (there are also the $\mathcal{X}$-sets of Shiota, among others). Rather than spell out the precise definition, let me roughly describe an o-minimal structure as consisting of subsets of $\mathbb{R}^n$ (where $n = 0, 1, 2, \ldots$) which

Are closed under all first-order logical operations: unions, intersections, relative complements, cartesian products, and closed under taking direct images along coordinate projections $\mathbb{R}^{n+1} \to \mathbb{R}^n$ (in order to accommodate existential quantification);

Include the equality and inequality relations $\{(x, y) \in \mathbb{R}^2: x = y\}$ and $\{(x, y) \in \mathbb{R}^2: x < y\}$;

*Do not include* any subsets of $\mathbb{R}^1$ except for arbitrary finite unions of points and intervals $(a, b)$, allowing $a = -\infty$ and $b = \infty$. In other words, that contain only those subsets of $\mathbb{R}$ which have to be there by the preceding axioms, and no more. This is the famous order-minimality or o-minimality condition; it rules out for example the set of natural numbers as belonging to an o-minimal structure.

Regarding this last axiom: logicians know how to exploit the arithmetic of natural numbers to define all sorts of chaotic and pathological structures. For example, if graphs of polynomial functions are admitted, and if $\mathbb{N} \subseteq \mathbb{R}$ is also admitted, then chaos ensues: a logician can write down some complicated finite formula in these predicates to define any Borel set you jolly well please, or for that matter any set in the projective hierarchy. Thus, the o-minimality axiom is there to ensure a level of respectable tameness.

The archetypal example of an o-minimal structure is the class of semi-algebraic sets, but many others are known. In any o-minimal structure, the "definable sets" (i.e., the sets belonging to the structure) admit Whitney stratifications into definable manifolds, and quite a rich theory has been developed; I refer to the book by van den Dries for details.

It would be very interesting if model theorists had looked into this business of the Bing-Borsuk conjecture; it could be that adding in o-minimality hypotheses would help address technical difficulties people have experienced in trying to prove this (but I am hardly qualified to say anything about this). A quick Google search didn't turn up much that I could see, but a paper by Frank Quinn in these proceedings seemed to touch ever so briefly on such issues.

Come to think of it, there are some resident experts on o-minimal theory here at MO (Thierry Zell and Dave Marker come to mind), and it would be wonderful if they could weigh in here.

In some sense, the quest for a definition of a stratified space is the same as the quest for the definition of a space of "finite topological type"."In fact, giving a workable definition [of stratified spaces] was historically a very difficult problem, and it is very possible that the definitions we have today will not turn out to be the best ones." $\endgroup$ – Junkie Jun 11 '11 at 0:3110more comments