Difference between revisions of "N-cell"

Latest revision as of 19:57, 14 January 2017

Definitions

Closed [ilmath]n[/ilmath]-cell

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

• Often [ilmath]X[/ilmath] will be a subset of another topological space and the topology will be the subspace topology [ilmath]X[/ilmath] inherits from it.

Open [ilmath]n[/ilmath]-cell

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

• Often [ilmath]X[/ilmath] will be a subset of another topological space and the topology will be the subspace topology [ilmath]X[/ilmath] inherits from it.

Characterisations

Caveat:There are (probably) other characterisations; like I should suspect any closed compact connected set with non-empty interior is a closed [ilmath]n[/ilmath]-cell

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

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

• [ilmath]X[/ilmath] is a closed [ilmath]n[/ilmath]-cell and its interior is an open [ilmath]n[/ilmath]-cell

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

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

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

Notes

1. ↑ The closed unit ball, often denoted: [ilmath]\overline{\mathbb{B}^n} [/ilmath] or [ilmath]\overline{\mathbb{B} }^n[/ilmath], is a closed ball of radius [ilmath]1[/ilmath] based at the origin.
   • [ilmath]\overline{\mathbb{B}^n}:\eq\{x\in\mathbb{R}^n\ \vert\ \Vert x\Vert\le 1\} [/ilmath], where the norm used is the standard Euclidean norm: [ilmath]\Vert x\Vert:\eq\sqrt{\sum^n_{i\eq 1} x_i^2} [/ilmath]
2. ↑ The open unit ball, denoted: [ilmath]\mathbb{B}^n [/ilmath], is an open ball of radius [ilmath]1[/ilmath] based at the origin.
   • [ilmath]\mathbb{B}^n:\eq\{x\in\mathbb{R}^n\ \vert\ \Vert x\Vert < 1\} [/ilmath], where the norm is the same as it is in the note for the closed unit ball
3. ↑ Considered with its usual topology. Given by the Euclidean norm of course
4. ↑ Recall the closed unit ball is:
   • [ilmath]\overline{\mathbb{B}^n}:\eq\{x\in\mathbb{R}^n\ \vert\ \Vert x\Vert\le 1\} [/ilmath] where the norm is the usual Euclidean norm:
     • [ilmath]\Vert x\Vert:\eq\sqrt{\sum^n_{i\eq 1}x_i^2} [/ilmath]
5. ↑ As before:
   • [ilmath]\mathbb{B}^n:\eq\{x\in\mathbb{R}^n\ \vert\ \Vert x\Vert<1\} [/ilmath], with the Euclidean norm mentioned in the note for closed unit ball above.

References

1. ↑ ^{1.0 1.1 1.2} Introduction to Topological Manifolds - John M. Lee