Topological manifold with boundary and related properties

Author: prabau

properties/P000235.md

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+---
+uid: P000235
+name: Locally a Euclidean half-space
+---
+
+Each point has a neighborhood homeomorphic to an open subset of the closed upper half-space $\mathbb R^n_+$ for some integer $n\ge 0$, where
+
+$\quad\quad\mathbb R^n_+:=\{(x_1,\dots,x_n)\in\mathbb R^n : x_n\ge 0\}$
+
+has the Euclidean subspace topology.
+Equivalently, each point has a neighborhood homeomorphic to $\mathbb R^n_+$ for some $n$.
+Note that the value of $n$ is allowed to differ between points;
+if it does not vary, then the space has the stronger
+property {P236}.
+
+In the case $n=0$, the half-space $\mathbb R^0_+$ is considered the same as $\mathbb R^0=\{0\}$;
+thus a point having a neighborhood homeomorphic to
+$\mathbb R^0_+$ means it is an isolated point.
+
+*Note*: Each point $p\in X$ for which $n>0$ falls into exactly one of two cases:
+- $p$ has a neighborhood homeomorphic to $\mathbb R^n$;
+- $p$ has a neighborhood homeomorphic to $\mathbb R^n_+$
+with $p$ mapped to a point of $\partial\mathbb R^n_+:=\{x\in\mathbb R^n: x_n=0\}$.
+
+The closely related {P236} property is the main ingredient used to define {P237}.

properties/P000236.md

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+---
+uid: P000236
+name: Locally an $n$-Euclidean half-space
+aliases:
+  - Locally a Euclidean half-space of dimension $n$
+refs:
+  - zb: "1209.57001"
+    name: Introduction to topological manifolds (Lee, 2011)
+  - wikipedia: Manifold_with_boundary
+    name: Manifold with boundary on Wikipedia
+---
+
+There exists some integer $n\ge 0$ such that each point of $X$ has a neighborhood
+homeomorphic to an open subset of the closed upper half-space $\mathbb R^n_+$, where
+
+$$\mathbb R^n_+:=\{(x_1,\dots,x_n)\in\mathbb R^n : x_n\ge 0\}$$
+
+has the Euclidean subspace topology.
+Equivalently, each point has a neighborhood homeomorphic to $\mathbb R^n_+$.
+Note that the value of $n$ is not allowed to differ between points;
+if it can vary, then the space has the weaker property {P235}.
+
+In the case $n=0$, the half-space $\mathbb R^0_+$ is considered the same as $\mathbb R^0=\{0\}$;
+thus a point having a neighborhood homeomorphic to
+$\mathbb R^0_+$ means it is an isolated point.
+
+*Note*: For $n>0$, each point $p\in X$ falls into exactly one of two cases:
+- ("manifold interior" point) $p$ has a neighborhood homeomorphic to $\mathbb R^n$;
+- ("manifold boundary" point) $p$ has a neighborhood homeomorphic to $\mathbb R^n_+$
+with $p$ mapped to a point of $\partial\mathbb R^n_+:=\{x\in\mathbb R^n: x_n=0\}$.
+
+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}}.

properties/P000237.md

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+---
+uid: P000237
+name: Topological $n$-manifold with boundary
+aliases:
+- $n$-dimensional topological manifold with boundary
+refs:
+  - zb: "1209.57001"
+    name: Introduction to topological manifolds (Lee, 2011)
+  - wikipedia: Manifold_with_boundary
+    name: Manifold with boundary on Wikipedia
+---
+
+A {P236} space that is {P3} and {P27}.
+
+Defined on page 42 of {{zb:1209.57001}}. See also {{wikipedia:Manifold_with_boundary}}.