I am trying to generalize the construction which takes a $p-$simplex $\phi$ to a $(p+1)-$simplex, $\theta$. But instead of letting $\phi$ be defined on $\sigma_p = \{(x_0,...,x_p):\Sigma \ x_i = 1,x_i \geq 0\}$ We let $\phi$ be defined on the set $\{(x_0,...,x_p):\Sigma \ x_i = 1\}$. And I want $\theta$ to be defined on $\{(x_0,...,x_{p+1}):\Sigma \ x_i = 1\}$. I want to do this to try to prove that if we replace the simplices $\Delta^n$ in the definition of homology with the spaces $\Xi^n=\{(x_0,...,x_p):\Sigma \ x_i = 1\}$ with the obvious inclusion maps you get similar or isomorphic theories.
I ran into a bit of a problem however, the proof of this fact relies on the compactness of $\Delta^n$ which $\Xi^n$ does not possess. Here is the proof of the continuity of $\theta$.
This proof does not work for my case since $\phi$ could grow too fast so we have to replace the factor $(1-t_0)$ in front of $\phi(...)$ with something that shrinks way faster but I have no idea how to to this.