📂Topological Data Analysis

Definition of CW Complex

Overview ¹

CW complexes are a type of complex also known as Cell Complexes, constructed through the following recursive procedure.

Definition

1. A discrete set $X^{0} \ne \emptyset$ is considered as a $0$-cell.
2. A $n$-skeleton $X^{n}$ is created by attaching $n$-cells $e_{\alpha}^{n}$ into $\phi_{\alpha} : S^{n-1} \to X^{n-1}$ from $X^{n-1}$.
3. When $X := \bigcup_{n \in \mathbb{N}} X^{n}$ becomes a topological space with a weak topology, $X$ is called a cell complex.

Explanation

Though the definition may seem difficult and complex, it is more approachable than it appears. Honestly, one does not need to know in great detail about CW complexes, so don’t feel too pressured.

Generalization of Graphs

The $1$-skeleton is a graph in itself. Here, the $0$-cells $X^{0} = V$ represent a set of vertices, and the $1$-cells $X^{1} = E$ represent a set of edges. $e_{\alpha}^{1} \in E$ is an edge that connects $0$-cells according to an index $\alpha$, and not necessarily all $0$-cells need to be connected.

From this perspective, it is acceptable to consider cell complexes as a generalization of graphs, known as Hyper Graphs. Below is an illustration of a hypergraph, where the edges $e_{k}$ that connect multiple vertices simultaneously correspond to the cells $e_{\alpha}$².

Origin of CW ³

Most literature does not use the term Cell Complex but commonly refers to them as CW complexes. Let’s delve into the definition to understand why.

Definition of Disks, Spheres, and Cells:

1. A $D^{n} \subset \mathbb{R}^{n}$ defined as follows is called a $n$-Unit Disk: $$ D^{n} := \left\{ \mathbf{x} \in \mathbb{R}^{n} : \left\| \mathbf{x} \right\| \le 1 \right\} $$
2. A $S^{n} \subset \mathbb{R}^{n+1}$ defined as follows is called a $n$-Unit Sphere: $$ S^{n} := \left\{ \mathbf{x} \in \mathbb{R}^{n+1} : \left\| \mathbf{x} \right\| = 1 \right\} $$
3. A $D^{n} \setminus \partial D^{n}$ and its homeomorphic subset $e^{n}$ are also called a $n$-Cell.

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

There should be no big issue with $X^{0}$. That a $e_{\alpha}^{n}$ attaches to a $n-1$ skeleton is similar to how a generalized $k$-edge in a hypergraph connects multiple vertices. To be more precise, we are creating a quotient space by assigning an equivalence relation $$ x \sim \phi_{\alpha} (x) $$ to all points $x \in \partial D_{\alpha}^{n}$ of the boundary through the map $\phi_{\alpha} : S^{n-1} \to X^{n-1}$ corresponding to the $n$-cell $e_{\alpha}^{n}$. If this explanation is difficult, one might need to review undergraduate-level topology. The idea of applying an equivalence relation in topology - treating different elements as essentially the same - intuitively means ‘connecting spaces together’. The $n$-skeleton $X^{n} := x^{n-1} \cup \bigsqcup_{\alpha} D_{\alpha}^{n}$ under these quotient mappings $\phi_{\alpha}$ is a quotient space, and $n$-cell $e_{\alpha}^{n}$ is homeomorphic to the image of $D_{\alpha}^{n} \setminus \partial D_{\alpha}^{n}$ under the quotient mapping $\phi_{\alpha}$.

Meanwhile, the fact that $X$ has a weak topology means that $A \subset X$ being open (closed) in $X$ is equivalent to $A \cap X^{n}$ being open (closed) in $X^{n}$ for all $n \in \mathbb{N}$. There’s no need to strenuously relate this to the general concept of weak topology; if it’s unclear, it’s fine to just move on.

After all, the CW complex got its name from the following two characteristics:

• Closure-finiteness: The closure of each cell intersects with only a finite number of other cells. This is related to compactness.
• Weak topology: The definition guarantees the presence of a weak topology.