📂Topological Data Analysis

Topology in Mathematics: Discs and Spheres

Definition ¹

In the Euclidean space $\left( \mathbb{R}^{n} , \left\| \cdot \right\| \right)$, the following shapes are defined.

1. Defined as $D^{n} \subset \mathbb{R}^{n}$, this is called $n$-Unit Disk. $$ D^{n} := \left\{ \mathbf{x} \in \mathbb{R}^{n} : \left\| \mathbf{x} \right\| \le 1 \right\} $$
2. Defined as $S^{n} \subset \mathbb{R}^{n+1}$, this is called $n$-Unit Sphere. $$ S^{n} := \left\{ \mathbf{x} \in \mathbb{R}^{n+1} : \left\| \mathbf{x} \right\| = 1 \right\} $$
3. An open subset $e^{n}$ that is homeomorphic to $D^{n} \setminus \partial D^{n}$ is also referred to as $n$-Cell.

Properties

The boundary of a $n$-disk is a $n$-sphere. In other words, the following holds true. $$ \partial D^{n} = S^{n-1} $$

Explanation

Disks and spheres are easier to understand when viewed from the perspective of $n=2$. In terms of representation, a $2$-disk is like the disks we encounter in daily life, a completely filled circular plate, while the $2$-sphere represents just the surface of a sphere at a higher $2+1$ dimension, without volume but with surface area.

As defined, $D^{3}$, though it does not look like a disk, is indeed a disk. Meanwhile, a cell, as seen, is defined through homeomorphism, so it does not necessarily need to be accurately defined as a set like disks and spheres.

This can be understood to mean that cells are free in terms of shape, size, and position, and by thinking about cells in this way, the widely known appearance of topology is revealed.

When $n=0$

When it’s $n = 0$, it is $D^{0} = \left\{ 0 \right\}$, and $e^{n}$ consists of a single point that is homeomorphic to it, but $S^{0}$ immediately implies that it has two points.

See Also

General Definition of a Sphere

A general sphere can be more mathematically defined through an inner product, and indeed, it can be easily generalized to an ellipsoid as well. However, disks and spheres are most commonly mentioned in topology, where specific coordinates or geometric properties are not strictly necessary.