Fix the details of a proof which will be referred to in CompHaus proofs

Author: dschepler

databases/catdat/data/003_category-property-assignments/Haus.sql

@@ -81,7 +81,7 @@ VALUES
 	'Haus',
 	'filtered-colimit-stable monomorphisms',
 	FALSE,
-	'The proof is similar to <a href="/category/Met">$\Met$</a>. For $n \geq 1$ let $X_n$ be the pushout of $[-1/n,+1/n] \hookrightarrow \IR$ with itself. That is, $X_n$ is the union of two lines $\IR \times \{1\}$ and $\IR \times \{2\}$ where we identify $(x,1) \equiv (x,2)$ when $|x| \leq 1/n$. Then $X_n$ is Hausdorff, and there is a canonical surjective continuous map $X_n \to X_{n+1}$. The colimit in $\Top$ is the union of two lines where we identify $(x,1) \equiv (x,2)$ when $|x| \leq 1/n$ for some $n$, i.e. when $x \neq 0$. This is the line with the double origin, which is not Hausdorff. Its Hausdorff reflection is the line $\IR$ where all points of both lines are identified, and it provides the colimit in $\Haus$. Now, the injective continuous maps $\{1,2\} \to X_n$, $i \mapsto (0,i)$ (where $\{1,2\}$ is discrete) become the constant map $0 : \{1,2\} \to \IR$ in the colimit, which is no monomorphism.'
+	'The proof is similar to <a href="/category/Met">$\Met$</a>. For $n \geq 1$ let $X_n$ be the pushout of $(-\infty, -1/n] \cup [1/n, \infty) \hookrightarrow \IR$ with itself. That is, $X_n$ is the union of two lines $\IR \times \{1\}$ and $\IR \times \{2\}$ where we identify $(x,1) \equiv (x,2)$ when $|x| \geq 1/n$. Then $X_n$ is Hausdorff, and there is a canonical surjective continuous map $X_n \to X_{n+1}$. The colimit in $\Top$ is the union of two lines where we identify $(x,1) \equiv (x,2)$ when $|x| \geq 1/n$ for some $n$, i.e. when $x \neq 0$. This is the line with the double origin, which is not Hausdorff. Its Hausdorff reflection is the line $\IR$ where all points of both lines are identified, and it provides the colimit in $\Haus$. Now, the injective continuous maps $\{1,2\} \to X_n$, $i \mapsto (0,i)$ (where $\{1,2\}$ is discrete) become the constant map $0 : \{1,2\} \to \IR$ in the colimit, which is not a monomorphism.'
 ),
 (
 	'Haus',