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# Cell

Unlike cells in a cubical complexes, the cells in cell complexes are closed.

We define the $n$-dimensional cell, or simply $n$-cell, as a topological space $C$ homeomorphic to the closed ball ${\bf B}^n$ in ${\bf R}^n$:

$$C \approx {\bf B}^n = \{x \in {\bf R}^n: ||x|| \leq 1 \}.$$

Then, a $0$-cell is simply a point. A $1$-cell is homeomorphic to a closed interval. A $2$-cell is homeomorphic to a disk.

As a subspace of ${\bf R}^n$, ${\bf B}^n$ is partitioned into the interior and frontier:

$${\rm Int}({\bf B}^n) = \{x \in {\bf R}^n: ||x|| < 1 \},$$

$${\rm Fr}({\bf B}^n) = \{x \in {\bf R}^n: ||x|| = 1 \}.$$

It follows then that the cell $C$ is also partitioned into the interior and frontier:

$${\rm Int}(C) \approx {\bf R}^n,$$

$${\rm Fr}(C) \approx {\bf S}^{n-1}.$$

The latter is more commonly called the boundary of the cell and denoted by $\partial C$. Keep in mind that the same notation is used for the boundary of a cell in a cell complex (or cubical complex), in the algebraic sense.