Alternate definition of singular homology. Instead of defining the spaces $\Delta^n=\{(x_0,...x_{n}) \in \mathbb R^{n+1}:\Sigma_i x_i=1 \ \text{&amp;} \ x_m \geq0\ \forall m\}$ define $s\Delta^n =\{(x_0,...x_{n}) \in \mathbb R^{n+1}:\Sigma_i x_i=1\}$ with the face maps given by $d_i:(x_0,..x_n) \mapsto (x_0,...x_{i-1},0,x_i...,x_n)$. Does this define an isomorphic version of homology? As in if X is a space, do the two definitions coincide. Not necessarily naturally.

I've been struggling with this problem for over two weeks now. I could elaborate on methods I have tried but none have been succesful so I don't see the point. I really only need the result for open subsets of $\mathbb R^n$ so make any assumptions you want about the space X. Thank you in advance!
Yes, this is naturally isomorphic to singular homology.  To prove it, first construct a sequence of retractions $r_n:s\Delta^n
\to\Delta^n$ which are compatible with the face maps.  For instance, $r_1$ will take the line $s\Delta^1$ and contract the ends of it down to just the bounded interval $\Delta^1$.  Then $r_2$ will contract the plane $s\Delta^2$ down to the triangle $\Delta^2$, contracting each line bounding the triangle down to the edge of the triangle according to $r_1$.  In general, having constructed $r_i$ for $i&lt;n$, the existence of such an $r_n$ follows from the Tietze extension theorem (or more geometrically, from obstruction theory, if you triangulate $s\Delta^n$ such that each of the faces are subcomplexes).

These maps $r_n$ will induce a map of chain complexes $r^*:C_*(X)\to sC_*(X)$ where $C_*(X)$ is the ordinary singular chain complex and $sC_*(X)$ is the singular chain complex using $s\Delta^n$ instead of $\Delta^n$.  There is also a map $i^*:sC_*(X)\to C_*(X)$ given simply by restricting from $s\Delta^n$ to $\Delta^n$, and $i^*r^*=1$.  To prove that $i^*$ induces isomorphisms on homology, it thus suffices to show that $r^*i^*:sC_*(X)\to sC_*(X)$ is chain-homotopic to the identity.

To construct the chain homotopy, we just extend the $r_n$'s to a sort of deformation retraction.  Specifically, let $P^n$ be the space obtained by gluing together $n$ copies of $s\Delta^n$ the same way that $n$ copies of $\Delta^n$ are glued together to form the standard triangulation of $\Delta^n\times[0,1]$.  In particular, $P^n$ has two "start and end" faces which correspond to the faces $\Delta^n\times\{0\}$ and $\Delta^n \times\{1\}$ of $\Delta^n\times[0,1]$, and "side" copies of $P^{n-1}$ which correspond to $F\times[0,1]$ for each face $F$ of $\Delta^n$.  We can now construct a sequence of maps $h_n:P^{n+1}\to s\Delta^n$ such that $h_n$ is given by $h_{n-1}$ on each side face (mapping to the corresponding face of $s\Delta^n$), is given by the identity on the bottom face, and is given by $r_n$ on the top face.  Indeed, given $h_i$ for each $i&lt;n$, the existence of $h_n$ with the required properties follows from the Tietze extension theorem.

These maps $h_n$ then induce maps $h_n^*:sC_n(X)\to sC_{n+1}(X)$, sending each simplex to the alternating sum of the $n+1$ simplices of $P^{n+1}$ which are mapped into $X$ by composing with $h_n$.  Since $h_n$ is given by $h_{n-1}$ on the side faces, these maps $h_n^*$ will define a chain homotopy between what the $h_n$'s do on the bottom and top faces, i.e. between the identity and $r^*i^*$.



More generally, similar arguments show that if you replace the simplices $\Delta^n$ with any sufficiently nice levelwise contractible cosimplicial space, the homology you get will be naturally isomorphic to singular homology (or more strongly, the "singular set" you get by mapping out of your cosimplicial space will be naturally weak equivalent to the usual singular set).  Here "sufficiently nice" turns out to mean "Reedy cofibrant"; more generally, if I'm not mistaken, then the following is true:


  Let $C$ be a model category and $A$ be a fibrant object in $C$.  Define $h_A:C^{\Delta}\to(\mathtt{Set}^{\Delta^{op}})^{op}$ by $h_A(X)_n=\operatorname{Hom}(X_n,A)$.  Then $h_A$ is a left Quillen functor for the Reedy model structure on $C^{\Delta}$, and in particular it preserves weak equivalences between Reedy cofibrant objects.


In other words, working very generally in a model category, you could define a "singular homology" theory using maps out of a cosimplicial object, and then the singular homology theories given by weak equivalent Reedy cofibrant cosimplicial objects agree on fibrant objects.
Your question is answered by looking for places where compactness of $\Delta^n$ is applied in singular homology theory. For example, the proof of the excision theorem uses compactness of $\Delta^n$ to guarantee that a certain interated barycentric subdivision operation stops after finitely many steps. 

Your affine space $s\Delta^n$ is not compact, so this kind of finiteness argument wouldn't work. In fact, its not at all clear to me how you would even define the barycentric subdivision of $s\Delta^n$.