T536 changes

theorems/T000536.md

@@ -7,7 +7,5 @@ if:
 then:
  P000123: true
 ---
-If a neighborhood homeomorphic to $\mathbb R^n$ and a neighborhood homeomorphic
-to $\mathbb R^m$ intersect, their intersection must be homeomorphic to open sets
-in both $\mathbb R^n$ and $\mathbb R^m$. Since $\mathbb R^n=\mathbb R^m$ only if
-$n=m$, any connected component of a {P122} space must be of a single dimension.
+
+If a point $x$ has at the same time an open neighborhood homeomorphic to $\mathbb R^n$ and an open neighborhood homeomorphic to $\mathbb R^m$ with $n\ne m$, their intersection would be homeomorphic to an open set in $\mathbb R^n$ and to an open set in $\mathbb R^m$. But this is not possible by invariance of domain. So the dimension $n$ at the point $x$ is completely determined by the point, and by definition of {P122} the set of points of a given dimension $n$ is open in $X$. Since $X$ is {P36}, the dimension must be the same at every point.