📂Topological Data Analysis

Definition of Compactifiable Spaces

Definition ^{1 2}

Homotopy Type: For two topological spaces $X, Y$, if there exist continuous functions $f : X \to Y$, $g: Y \to X$ satisfying the following, then $X, Y$ are said to have the same Homotopy Type. $X, Y$ or $f, g$ is also referred to as Homotopy Equivalence. $$\begin{align*} g \circ f \simeq& \text{id}_{X} \\ f \circ g \simeq& \text{id}_{Y} \end{align*}$$ Here, $\text{id}_{\cdot}$ is the identity function, and $f \simeq g$ signifies that $f,g$ is homotopic.

A topological space $X$ is considered a Contractible Space if it is homotopy equivalent to a space consisting of a single point $\left\{ x \right\} \subset X$.

Examples

No matter how vast the universe is, what significance does it hold when considering the multiverse, and what meaning does the volume of a particle with mass have if it is practically as small as a point?

In topology, the concepts of ‘coming to a point’ or ‘changing shape’ are extremely important, and intuitively, a contractible space could be described as a space that ‘cannot be made any smaller’, ’to a point’, ‘where it is permissible to think of changing its shape.’

Euclidean Space

In a Euclidean space $\mathbb{R}^{p}$, imagine any path and they can continuously transform into a point. That is, they are homotopic to the constant path $c_{x}$, hence, the Euclidean space is a contractible space.

$n$-Disk

A Disk $D^{n}$, being even smaller than a Euclidean space, is obviously contractible.

Convex Sets

Even if not to the extent of a disk, it’s evident that a convex hull is trivially contractible as a subset of Euclidean space.

The Unit Circle is Not Contractible

On the other hand, the unit circle $S^{1}$, having a hole in the middle, cannot be contracted to a point. This fact is especially linked to why (beyond undergraduate level) topology is so obsessed with the concepts of ’loops’ and ‘holes’, together with homotopy and fundamental group.