Definable homeomorphisms in o-minimal geometry I am currently working through the proof of the so called "triangulation theorem" on page 130 in Lou van den Dries book "Tame Topology and o-minimal Structures.". 

Triangulation Theorem: Let $S\subset R^m$ be a definable set, with definable subsets $S_1,...,S_k$. Then $S$ has a triangulation in $R^m$ that is compatible with these subsets.

I got ruminative at the following passage: 

"Therefore we modify $F$ as follows to $\tilde{F}$. Each $f\in F$ extends, first continuously to the
  closure of its domain, and then, by the last remark of (2.3), further to a continuous definable function $\tilde{f}:A\rightarrow R$."

I went on and looked up the remark (2.3), which, in my opinion, demands the $\Phi$ function in the definition of a triangulation to be continuous in both directions (i.e. a topological homeomorphism), in order to fullfill. 
The full "remark" (2.3) in vdD book is:

(2.3) Definition. Let $A\subset R^m$ be a definable set. A triangulation in $R^n$ of $A$ is a pair $(\Phi,K)$ consisting of a complex $K$ in $R^n$ and a definable homeomorphism $\Phi:A\rightarrow \mid K \mid$. Note that then $\Phi^{-1}(K):=\left\{ \Phi^{-1}(\sigma): \sigma\in K\right\}$ is a finite partition of A. We call $(A,\Phi^{-1}(K))$ a triangulated set. The triangulation is said to be compatible with the subset $A'\subset A$ if $A'$ is a union of elements of $\Phi^{-1}(K)$.
  Note that by Chapter 6.(1.10), we have: A is closed and bounded $\iff$ the complex K is closed (where $A\subset R^m$ is a definable set). In that case each continuous definable R-valued function on $cl(C)$, where $C\in\Phi^{-1}(K)$, has a continuous definable R-valued extension to A, by lemma (2.2).

Where chapter 6, (1.10) says:

(1.10) Proposition. If $f:X\rightarrow R^n$ is a continuous definable map on a closed bounded set $X\subset R^m$, then $f(X)$ is closed and bounded in $R^n$.

and lemma (2.2) says

(2.2) Lemma. Let $K$ be a closed complex in $R^m$ and $L$ a closed subcomplex. Then each continuous definable function $f:\mid L \mid \rightarrow R$ has a continuous definable extension $\tilde{f}:\mid K \mid \rightarrow R$.

So, in order to apply lemma (2.2) we need a closed simplicial complex $K$, which follows from the continuity of $\Phi$ (here we need the continuity) and proposition (1.10). Since in the proof of the triangulation theorem, the functions $f$ which ought to be extended to $A$ have sets of $\Phi^{-1}(K)$ as a domain, we need to consider $f\circ \Phi$ in the situation of lemma (2.2), which also needs $\Phi$ to be continuous.
I am confused a bit now because this paper calls the "definable homeomorphisms" in the triangulation theorem "definable bijections", hence not necessarily continuous (referring to the attribute of a homeomorphism as structure preserving in case of the "definable" structure).
Page 4 says: 

A definable homeomorphism is a tame bijection between tame sets.  To repeat: definable homeomorphisms are not necessarily continuous.

When you want to define the so called definable Euler characteristic  like on page 4 here

Definition 2.2. If $X\in\mathcal{O}$ is tame and $h:X\rightarrow \cup \sigma_i$ is a definable bijection with a collection of open simplices, then the definable Euler characteristic of X is: $\chi(X):=\sum_{i}(-1)^{dim\;\sigma_i}$, where $dim\;\sigma_i$ is the dimension of the open simplex $\sigma_i$. We understand that $\chi(\emptyset)=0$ since this corresponds to the empty sum.

where, in my opinion, the definable bijection term arises from the understanding of a definable homeomorphism as a definable bijection, because in the same paper the triangulation theorem from van der Dries book is cited as:

Theorem 2.1 (Triangulation Theorem [16]). Any tame set admits a definable bijection with a subcollection of open simplices in the  geometric realization of a finite Euclidean simplicial complex. Moreover, this bijection can be made to respect a partition of a tame set into tame subsets.

Although page 70 of van den Dries book basically says a definable injection suffices for the well-definedness of the Euler characteristic

(2.4) Proposition: If $f:S\rightarrow R^n$ is an injective definable map, then $E(S)=E(f(S))$. 

a definable homeomorphism in sense of a definable, bijective, continuous function with continuous inverse, will make life in case of the well-definedness of the definable Euler characteristic much easier, because there wont be any need to refer to (2.4) of van den Dries book. 


I browsed van den Dries' book a bit, but i didn't find the definition of a "definable homeomorphisms". So do they need to be definable, bijective and continous with a continuous inverse or do they only need to be definable and bijective (which leads to a problem with the remark (2.3) above).
Thank you in advance, 
Soucerer
 A: Answering your last question first: A definable homeomorphism is a function which is both definable and a homeomorphism. So yes, a definable homeomorphism (between subsets of an o-minimal structure) is continuous (in the o-minimal topology) and has a continuous inverse, and is therefore bijective. 
I would be very surprised if the term "definable homeomorphism" were ever used in a model theory context for anything other than a continuous definable function with continuous inverse, in some setting with a natural topological structure. The topological component also seems crucial to the concept of triangulation, so I'm surprised that you found a paper that defines triangulation in terms of maps that are only definable bijections. Are you sure you've understood that paper correctly?
I looked up the reference to (2.3) in van den Dries's book. He is referring to Definition 2.3 on p. 127, and specifically to the remark that occurs at the end of the definition, at the top of p. 128. It reads:

In that case, each continuous definable $R$-valued function on $\text{cl}(C)$, where $C\in \Phi^{-1}(K)$, has a continuous definable $R$-valued extension to $A$, by lemma (2.2). 

I don't see here any requirement that any function be a homeomorphism. So it does seem to apply to the situation you quote in your question, which is about extending a continuous function on the closure of a set up to a continuous function on $A$. 

Added, in response to further comments by the OP. 
With all due respect to Curry, Ghrist, and Robinson, the way they use the term "definable homeomorphism" is definitely nonstandard, and it seems to me like a really bad idea. For one thing, the word homeomorphism gives entirely the wrong intuition. And for another thing, we already have a perfectly good term: definable bijection. 
Now it turns out that in an o-minimal context, every definable bijection $f$ is a piecewise homeomorphism: you can partition the domain and codomain each into finitely many pieces such that $f$ restricts to a homeomorphism between pieces. So maybe Curry, Ghrist, and Robinson want to emphasize this behavior, and suggest that a definable bijection is actually much tamer than an arbitrary bijection. But in this case, it would be better to use van den Dries's term "definable equivalence" (see (2.11) on p. 132) or at least say "definable piecewise homeomorphism". 
Again, van den Dries definitely means "definable and continuous with continuous inverse" when he writes "definable homeomorphism", and this is the standard meaning of the term. It seems to me that with this reading, your concerns about the proof of the theorem in van den Dries's book are resolved (and I agree with you that the passage in question requires $\Phi$ to be continuous).
