open nbhds

Author: prabau

properties/P000122.md

@@ -17,5 +17,7 @@ property {P123}.
 In the case that $n=0$, the point having a neighborhood homeomorphic to
 $\mathbb R^0=\{0\}$ means it is an isolated point.
 
+*Note*: In all the above, one can equivalently require the neighborhoods to be open.
+
 The closely related {P123} property is
 defined on page 316 of {{zb:0951.54001}} and on page 38 of {{mr:2766102}}.

properties/P000123.md

@@ -20,4 +20,6 @@ property {P122}.
 In the case that $n=0$, the point having a neighborhood homeomorphic to
 $\mathbb R^0=\{0\}$ means it is an isolated point.
 
+*Note*: In all the above, one can equivalently require the neighborhoods to be open.
+
 Defined on page 316 of {{zb:0951.54001}} and on page 38 of {{mr:2766102}}.

properties/P000235.md

@@ -23,4 +23,6 @@ one of the following holds:
 - $p$ has a neighborhood homeomorphic to $\mathbb R^n_+$
 with $p$ mapped to a point of $\partial\mathbb R^n_+:=\{x\in\mathbb R^n: x_n=0\}$.
 
+*Note*: In all the above, one can equivalently require the neighborhoods to be open.
+
 The closely related {P236} property is the main ingredient used to define {P237}.

properties/P000236.md

@@ -29,6 +29,8 @@ Equivalently when $n>0$, each point $p\in X$ satisfies one of the following:
 - (*"manifold boundary" point*) $p$ has a neighborhood homeomorphic to $\mathbb R^n_+$
 with $p$ mapped to a point of $\partial\mathbb R^n_+:=\{x\in\mathbb R^n: x_n=0\}$.
 
+*Note*: In all the above, one can equivalently require the neighborhoods to be open.
+
 This property is the main ingredient in the definition of {P237}.
 
 Mentioned on page 42 of {{zb:1209.57001}}. See also {{wikipedia:Manifold_with_boundary}}.

theorems/T000847.md

@@ -1,10 +1,12 @@
 ---
 uid: T000847
 if:
-  P000122: true
+  P000235: true
 then:
   P000223: true
 ---
 
-For each $x\in X$, every neighborhood of $x$ contains an open neighborhood homeomorphic to some Euclidean space $\mathbb R^n$.
-And $\mathbb R^n$ is {P199} as it can be deformation retracted to a point using a straight-line homotopy.
+For each $x\in X$, every neighborhood of $x$ contains an open neighborhood homeomorphic to
+some Euclidean space $\mathbb R^n$ or some closed half-space $\mathbb R^n_+$.
+And $\mathbb R^n$ and $\mathbb R^n_+$ are both {P199}
+as they can be deformation retracted to a point using a straight-line homotopy.