$\Delta$-Complexes Are Hausdorff I am using the definition of a $\Delta$-complex as given in Hatcher's book  here on pg 103.
Now on pg. 104, just before the section on Simplicial Homology, Hatcher remarks that if $X$ has a $\Delta$-complex structure then $X$ is Hausdorff.
I do not see it at all. Is this easy? Does anybody know of a reference where the general topological details of a $\Delta$-comlpex can be found?
Thanks.
 A: I am going through Hatcher now, stopped at the same point as you, and had the same question. I think I have a proof.
Let $p,q$ be distinct in $X$. By the definition of a $\Delta$-complex, we have that $\sigma_a(p')=p$ and $\sigma_b(q')=q$ for exactly one $a,b$ not necessarily distinct. Here, $\sigma_a:\Delta^n\to X$ and $\sigma_b:\Delta^{n'} \to X$.
Suppose $a$ and $b$ are distinct. Then by condition (i) in the definition of a $\Delta$-complex in Hatcher, $p'$ and $q'$ lie in interiors of different simplices. These interiors are disjoint and open in $\mathbb{R}^n$,$\mathbb{R}^{n'}$. Because $\mathbb{R}^k$ is Hausdorff, $p',q'$ lie disjoint neighborhoods open in their respective open simplices. The images of these disjoint open neighborhoods under $\sigma_a,\sigma_b$ are disjoint because of condition (i) in tandem with their being disjoint. And the images are open in $X$ because of condition (iii) in tandem with their being open. In particular, $p,q$ have disjoint open neighborhoods.
Now suppose $a$ and $b$ are not distinct. Then $p',q'$ lie in the interior of an $n$-simplex. This is open in $\mathbb{R}^n$ which is Hausdorff, hence lie in disjoint neighborhoods open in the $n$-simplex. Then a more or less identical argument as before applies; the images under $\sigma_a$ of these open sets must be disjoint by (i), and must be open by (iii), hence we have found disjoint open neighborhoods of $p,q$.
I hope this is correct and helpful to whoever reads it. It does not seem like I actually needed to split this problem into cases.

As pointed out in comments, this proof is not correct. I will work on fixing it when I have time.
