Boundary (topology)

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Deprecate Closure, interior and boundary

Definition

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

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

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

Equivalent definition

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Proof would be good, check, both neighbourhoods

[ilmath]p\in\partial A\iff\forall U\in\mathcal{J}[p\in U\implies(U\cap A\neq\emptyset\wedge U\cap(X-A)\neq\emptyset)][/ilmath]^[1]
- I think this can be relaxed to our definition of a neighbourhood though.

Notes

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

References

↑ ^1.0 ^1.1 Introduction to Topological Manifolds - John M. Lee

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

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

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

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

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

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

[ITTMJML-1] 1.0 ^1.1 Introduction to Topological Manifolds - John M. Lee

[1]

[Note 1]

Boundary (topology)

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Definition

Equivalent definition

Notes

References

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