Boundary (topology)

Stub grade: A*

This page is a stub

This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:

Link to other pages then demote. Specifically:

Include in List of topological properties
Deprecate Closure, interior and boundary

Definition

Let $(X,\mathcal{ J })$ be a topological space and let $A\in\mathcal{P}(X)$ be an arbitrary subset of $X$ . Then the boundary of $A$ , denoted $\partial A$ is defined as^[1]:

$\partial A:\eq X-(\text{Int}(A)\cup\text{Ext}(A))$ - where $\text{Int}(A)$ denotes the interior of $A$ and $\text{Ext}(A)$ denotes the exterior of $A$ ^{[Note 1]}

A point $p\in\partial A$ is called a boundary point of $A$ . Caveat:There are many other uses for "boundary point" throughout mathematics

Equivalent definition

Grade: A*

This page requires some work to be carried out

Some aspect of this page is incomplete and work is required to finish it
The message provided is:

Proof would be good, check, both neighbourhoods

p∈∂A⟺∀U∈J[p∈U⟹(U∩A≠∅∧U∩(X−A)≠∅)]^[1]
- I think this can be relaxed to our definition of a neighbourhood though.

Notes

Jump up ↑
Stated for convenience:
1. $\text{Int}(A):\eq\bigcup\left\{U\in\mathcal{P}(X)\ \vert\ U\subseteq A\wedge U\in\mathcal{J} \right\}$ - recall $\mathcal{J}$ is the set of open sets of the topology, so $U\in\mathcal{J}\iff U$ is open in $(X,\mathcal{ J })$ .
2. $\text{Ext}(A):\eq X-\overline{A}$ - where $\overline{A}$ denotes the closure of $A$
3. $\overline{A}:\eq\bigcap\left\{C\in\mathcal{P}(X)\ \vert\ A\subseteq C\wedge C\text{ closed in }X\right\}$ - a set is closed if and only if its complement is open.

References

↑ ^{Jump up to: 1.0} ^1.1 Introduction to Topological Manifolds - John M. Lee

[2] Jump up ↑ Stated for convenience:
$\text{Int}(A):\eq\bigcup\left\{U\in\mathcal{P}(X)\ \vert\ U\subseteq A\wedge U\in\mathcal{J} \right\}$ - recall $\mathcal{J}$ is the set of open sets of the topology, so $U\in\mathcal{J}\iff U$ is open in $(X,\mathcal{ J })$ .

$\text{Ext}(A):\eq X-\overline{A}$ - where $\overline{A}$ denotes the closure of $A$

$\overline{A}:\eq\bigcap\left\{C\in\mathcal{P}(X)\ \vert\ A\subseteq C\wedge C\text{ closed in }X\right\}$ - a set is closed if and only if its complement is open.

[2] $\text{Int}(A):\eq\bigcup\left\{U\in\mathcal{P}(X)\ \vert\ U\subseteq A\wedge U\in\mathcal{J} \right\}$ - recall $\mathcal{J}$ is the set of open sets of the topology, so $U\in\mathcal{J}\iff U$ is open in $(X,\mathcal{ J })$ .

[3] $\text{Ext}(A):\eq X-\overline{A}$ - where $\overline{A}$ denotes the closure of $A$

[4] $\overline{A}:\eq\bigcap\left\{C\in\mathcal{P}(X)\ \vert\ A\subseteq C\wedge C\text{ closed in }X\right\}$ - a set is closed if and only if its complement is open.

[ITTMJML-1] {Jump up to: 1.0} ^1.1 Introduction to Topological Manifolds - John M. Lee

[1]

[Note 1]

Boundary (topology)

Contents

Definition

Equivalent definition

Notes

References

Navigation menu

Views

Personal tools

Navigation

Search

Tools