$n$ -cell

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Definitions

Closed $n$ -cell

A topological space, $(X,\mathcal{ J })$ , is a "closed $n$ -cell" if it is homeomorphic to the closed unit ball^{[Note 1]} in $\mathbb{R}^n$ ^[1].

Often $X$ will be a subset of another topological space and the topology will be the subspace topology $X$ inherits from it.

Open $n$ -cell

A topological space, $(X,\mathcal{ J })$ , is a "open $n$ -cell" if it is homeomorphic to the open unit ball^{[Note 2]} in $\mathbb{R}^n$ ^[1].

Often $X$ will be a subset of another topological space and the topology will be the subspace topology $X$ inherits from it.

Characterisations

like I should suspect any closed compact connected set with non-empty interior is a closed $n$ -cell

If $X\in\mathcal{P}(\mathbb{R}^n)$ is compact and convex in $\mathbb{R}^n$ with non-empty interior then it is a closed $n$ -cell and its interior is an open $n$ -cell

Let $X\in\mathcal{P}(\mathbb{R}^n)$ be an arbitrary subset of $\mathbb{R}^n$ ^{[Note 3]}, then, if $X$ is compact and convex, and has a non-empty interior then^[1]:

$X$ is a closed $n$ -cell and its interior is an open $n$ -cell

Furthermore, given any point $p\in\text{Int}(X)$ , there exists a homeomorphism, $f:\overline{\mathbb{B}^n}\rightarrow X$ (where $\overline{\mathbb{B}^n}$ is the closed unit ball^{[Note 4]} in $\mathbb{R}^n$ ) such that:

$f(0)\eq p$
$f\left(\mathbb{B}^n\right)\eq\text{Int}(X)$ (where $\mathbb{B}^n$ is the open unit ball^{[Note 5]} in $\mathbb{R}^n$ ), and
$f(\mathbb{S}^{n-1})\eq\partial X$ (where $\mathbb{S}^{n-1}\subset\mathbb{R}^n$ is the $(n-1)$ -sphere, and $\partial X$ denotes the boundary of $X$ )

Caveat:When we speak of interior and boundary here, we mean considered as a subset of $\mathbb{R}^n$ , not as $X$ itself against the subspace topology on $X$

Notes

Jump up ↑
The closed unit ball, often denoted: ¯Bn or ¯Bn, is a closed ball of radius 1 based at the origin.
- $\overline{\mathbb{B}^n}:\eq\{x\in\mathbb{R}^n\ \vert\ \Vert x\Vert\le 1\}$ , where the norm used is the standard Euclidean norm: $\Vert x\Vert:\eq\sqrt{\sum^n_{i\eq 1} x_i^2}$
Jump up ↑
The open unit ball, denoted: Bn, is an open ball of radius 1 based at the origin.
- $\mathbb{B}^n:\eq\{x\in\mathbb{R}^n\ \vert\ \Vert x\Vert < 1\}$ , where the norm is the same as it is in the note for the closed unit ball
Jump up ↑ Considered with its usual topology. Given by the Euclidean norm of course
Jump up ↑
Recall the closed unit ball is:
- ¯Bn:={x∈Rn | ∥x∥≤1} where the norm is the usual Euclidean norm:
  - $\Vert x\Vert:\eq\sqrt{\sum^n_{i\eq 1}x_i^2}$
Jump up ↑
As before:
- $\mathbb{B}^n:\eq\{x\in\mathbb{R}^n\ \vert\ \Vert x\Vert<1\}$ , with the Euclidean norm mentioned in the note for closed unit ball above.

References

↑ ^{Jump up to: 1.0} ^1.1 ^1.2 Introduction to Topological Manifolds - John M. Lee

[1] Jump up ↑ The closed unit ball, often denoted: $\overline{\mathbb{B}^n}$ or $\overline{\mathbb{B} }^n$ , is a closed ball of radius $1$ based at the origin.
$\overline{\mathbb{B}^n}:\eq\{x\in\mathbb{R}^n\ \vert\ \Vert x\Vert\le 1\}$ , where the norm used is the standard Euclidean norm: $\Vert x\Vert:\eq\sqrt{\sum^n_{i\eq 1} x_i^2}$

[2] $\overline{\mathbb{B}^n}:\eq\{x\in\mathbb{R}^n\ \vert\ \Vert x\Vert\le 1\}$ , where the norm used is the standard Euclidean norm: $\Vert x\Vert:\eq\sqrt{\sum^n_{i\eq 1} x_i^2}$

[3] Jump up ↑ The open unit ball, denoted: $\mathbb{B}^n$ , is an open ball of radius $1$ based at the origin.
$\mathbb{B}^n:\eq\{x\in\mathbb{R}^n\ \vert\ \Vert x\Vert < 1\}$ , where the norm is the same as it is in the note for the closed unit ball

[4] $\mathbb{B}^n:\eq\{x\in\mathbb{R}^n\ \vert\ \Vert x\Vert < 1\}$ , where the norm is the same as it is in the note for the closed unit ball

[4] Jump up ↑ Considered with its usual topology. Given by the Euclidean norm of course

[5] Jump up ↑ Recall the closed unit ball is:
$\overline{\mathbb{B}^n}:\eq\{x\in\mathbb{R}^n\ \vert\ \Vert x\Vert\le 1\}$ where the norm is the usual Euclidean norm:
$\Vert x\Vert:\eq\sqrt{\sum^n_{i\eq 1}x_i^2}$

[7] $\overline{\mathbb{B}^n}:\eq\{x\in\mathbb{R}^n\ \vert\ \Vert x\Vert\le 1\}$ where the norm is the usual Euclidean norm:
$\Vert x\Vert:\eq\sqrt{\sum^n_{i\eq 1}x_i^2}$

[8] $\Vert x\Vert:\eq\sqrt{\sum^n_{i\eq 1}x_i^2}$

[6] Jump up ↑ As before:
$\mathbb{B}^n:\eq\{x\in\mathbb{R}^n\ \vert\ \Vert x\Vert<1\}$ , with the Euclidean norm mentioned in the note for closed unit ball above.

[10] $\mathbb{B}^n:\eq\{x\in\mathbb{R}^n\ \vert\ \Vert x\Vert<1\}$ , with the Euclidean norm mentioned in the note for closed unit ball above.

[ITTMJML-2] {Jump up to: 1.0} ^1.1 ^1.2 Introduction to Topological Manifolds - John M. Lee

[Note 1]

[1]

[Note 2]

[Note 3]

[Note 4]

[Note 5]

$n$ -cell

Contents

Definitions

Closed $n$ -cell

Open $n$ -cell

Characterisations

If $X\in\mathcal{P}(\mathbb{R}^n)$ is compact and convex in $\mathbb{R}^n$ with non-empty interior then it is a closed $n$ -cell and its interior is an open $n$ -cell

Notes

References

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nn-cell

Contents

Definitions

Closed nn-cell

Open nn-cell

Characterisations

If X∈P(Rn)X\in\mathcal{P}(\mathbb{R}^n) is compact and convex in Rn\mathbb{R}^n with non-empty interior then it is a closed nn-cell and its interior is an open nn-cell

Notes

References

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$n$ -cell

Closed $n$ -cell

Open $n$ -cell

If $X\in\mathcal{P}(\mathbb{R}^n)$ is compact and convex in $\mathbb{R}^n$ with non-empty interior then it is a closed $n$ -cell and its interior is an open $n$ -cell