Invariance of combinatorial/geometric euler characteristic I am trying to read and understand the paper: 
TARGET ENUMERATION VIA EULER CHARACTERISTIC INTEGRALS 
by YULIY BARYSHNIKOV AND
ROBERT GHRIST.
And I am having trouble with a statement. First of all, definitions:

First remark The definition of $k$-simplex should be with $t_i \in (0,1]$.
Second remark  The above definition differs from the common one. 
Now we continue and we introduce my problem:

Remark It is clear it is not an homotopic invariant.
MY QUESTION(S):


*

*Is there another easier (which needs less machinery) way to prove the
topological invariance of this Euler Characteristic?

*I mean, I have seen that Borel-Moore Homology is defined using sheafs
or sheaves. I am still a masters student and I have never heard about
that. Is there a proof of the statement using machinery from a first
course in Algebraic Topology (Some homotopy theory, a bit of homology
and cohomology)?

*In the case some heavy machinery is needed could you provide me some
references to look at?


WHAT I HAVE TRIED:


*

*I have tried using Cohomology with compact supports since in Massey's
book titled Singular Homology Theory he uses it to deal with non
compact manifolds in  order to prove Poincarè Duality. He refers to 

H. Cartan, Seminaire Henri Cartan 1948/49: Topologie Algebrique

which I have partially read. However, since the spaces we are dealing with are not
locally-compact, I can't use that stuff.

*I have read in the paper:  T. Beke, “Topological invariance of the
combinatorial euler characteristic of tame spaces,” an idea about
using one-point compactifications ... but I didn't achieve my goal
yet.

*I have had a look at Stack Exchange. There is a related this
question. However, I haven't found an answer there. I quote the
part I find most related.

The definition of combinatorial Euler characteristic is great for
  "finite polyhedral complexes", I think. By a "finite polyhedral
  complexes" I mean glue together finitely many polyhedra, but you're
  allowed to leave some faces open, so that unlike a CW complex not
  every cell must have complex closure. Then you can calculate Euler
  characteristic with the usual formula: (number of cells of even
  dimension) - (number of cells of odd dimension). I think this is a
  topological (but not homotopy!) invariant.

So thanks in advance and any help will be appreciated. 
 A: *

*I do not see the point of introducing Borel-Moore homology here, but maybe it is used elsewhere in the paper, I do not know. For simplicial complexes, singular homology suffices. My suggestion is to stick to closed simplicial complexes. 

*For closed finite simplicial complexes, $\chi$ is a homotopy-invariant. In particular, if you have two closed finite simplicial complexes $X, Y$  whose geometric realizations are homeomorphic then $\chi(X)=\chi(Y)$. I do not think there is any easier proof of this fact than identifying $\chi$ with 
$$
\sum_{i} (-1)^i b_i(X)
$$
(using simplicial homology) and then proving that simplicial homology is isomorphic to the singular homology. Proving these facts is not hard but a bit tedious. You can find proofs in any algebraic topology textbook, for instance in Hatcher's "Algebraic Topology". 

*For the definition of a closed simplex, it is indeed $t_i\in [0,1]$, not $t_i\in (0,1)$. However, for some reason, they also want to work with open simplices, in which case indeed $t_i\in (0,1)$. Why is it needed in their paper I do not know. Their definition of a (finite) simplicial complex seems sloppy, I think, they are assuming that it is (linearly) embedded in $R^n$ for some $n$ in which case it is OK. For simplicial complexes in their sense, $\chi$ is not a homotopy invariant. One can see this as follows. Take $X=(0,1)$, an open 1-simplex. It is homotopy-equivalent to $[0,1)$. But their Euler characteristics (as defined in their paper) are different. 

*See also 
H.Delfs, "Homology of Locally Semialgebraic Spaces," 
which, if I remember it correctly, defines the "right" homology theory in the context of their simplicial complexes, which, I think, is simpler than Borel-Moore.   
