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.
+
+An equivalent condition for points $p\in X$ for which $n>0$ is that
+one of the following holds:
+- $p$ 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
+
+$\quad\quad\mathbb R^n_+:=\{(x_1,\dots,x_n)\in\mathbb R^n : x_n\ge 0\}$
+
+has the Euclidean subspace topology.
+Equivalently, each point has a neighborhood homeomorphic to $\mathbb R^n_+$.
+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.
+
+Equivalently when $n>0$, each point $p\in X$ satisfies one of the following:
+- (*"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
+---
+
+$X$ is {P236} and {P3} and {P27}.
+
+Defined on page 42 of {{zb:1209.57001}}. See also {{wikipedia:Manifold_with_boundary}}.

theorems/T000290.md

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+---
+uid: T000290
+if:
+  P000237: true
+then:
+  P000236: true
+---
+
+By definition.

theorems/T000316.md

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+---
+uid: T000316
+if:
+  and:
+    - P000236: true
+    - P000003: true
+    - P000027: true
+then:
+  P000237: true
+---
+
+By definition.

theorems/T000774.md

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+---
+uid: T000774
+if:
+  P000122: true
+then:
+  P000235: true
+---
+By definition.

theorems/T000775.md

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+---
+uid: T000775
+if:
+  P000123: true
+then:
+  P000236: true
+---
+
+By definition.

theorems/T000776.md

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+---
+uid: T000776
+if:
+  P000236: true
+then:
+  P000235: true
+---
+
+By definition.