Lecture 9. Cell Complexes and Basic Cells
← Back to Course HomeKSM3E02: Algebraic Topology I, Fall 2026 | Wednesday, September 9, 2026 (12:00–13:00) | Instructor: Pranav Haridas
This lecture introduces cell complexes. We study whether a topological space can be decomposed into tractable pieces or assembled from elementary building blocks.
1. Decompositions and Building Blocks
Let $X$ be a topological space. We consider two dual questions. Can $X$ be decomposed into simpler subspaces that are tractable? Conversely, can spaces be constructed systematically from standardized building blocks?
Guiding Question
Can a topological space be broken down into tractable pieces, or built from elementary building blocks?
Euclidean spaces provide concrete models for this idea.
- The real line $\mathbf{R}$ decomposes into discrete points and open or closed intervals.
- The plane $\mathbf{R}^2$ can be assembled from points, line segments, and rectangles or closed disks glued along their boundaries.
In each dimension, the space decomposes into disjoint open cells. The boundary of each higher-dimensional cell is glued to cells of lower dimension.
Cell Dimension Progression
$0$-cells are points. $1$-cells are open intervals homeomorphic to $\mathbf{B}^1$. $2$-cells are open squares homeomorphic to $\mathbf{B}^2$.
Cell Decompositions in $\mathbf{R}$ and $\mathbf{R}^2$
Select a space to view its decomposition into elementary cells.
Figure 1. Decomposing $\mathbf{R}$ into $0$-cells $\{n\}$ and $1$-cells $[n, n+1]$.
We formalize this procedure for general spaces by defining cells.
2. Open and Closed $n$-Cells
Let $\mathbf{B}^n = \{ x \in \mathbf{R}^n : |x| < 1 \}$ denote the open unit ball in $\mathbf{R}^n$. Let $\overline{\mathbf{B}}^n = \{ x \in \mathbf{R}^n : |x| \le 1 \}$ denote the closed unit ball. The boundary sphere is $\mathbf{S}^{n-1} = \partial \overline{\mathbf{B}}^n = \{ x \in \mathbf{R}^n : |x| = 1 \}$.
Definition (Open and Closed Cells)
- An open $n$-cell is a topological space homeomorphic to $\mathbf{B}^n$.
- A closed $n$-cell is a topological space homeomorphic to $\overline{\mathbf{B}}^n$.
For $n = 0$, the spaces $\mathbf{B}^0$ and $\overline{\mathbf{B}}^0$ are single points. A singleton space is therefore both an open $0$-cell and a closed $0$-cell.
Open and closed balls in $\mathbf{R}^n$ are open and closed cells. Many compact convex sets are also cells.
3. Compact Convex Subsets
Proposition 5.1 (Lee, Chapter 5)
Let $D \subseteq \mathbf{R}^n$ be a compact convex subset with non-empty interior. Then $D$ is a closed $n$-cell and its interior $\operatorname{Int} D$ is an open $n$-cell.
Given any point $p \in \operatorname{Int} D$, there exists a homeomorphism $$ F: \overline{\mathbf{B}}^n \to D $$ that sends $0$ to $p$, $\mathbf{B}^n$ to $\operatorname{Int} D$, and $\mathbf{S}^{n-1}$ to $\partial D$.
Proof
Let $p \in \operatorname{Int} D$. The translation $x \mapsto x - p$ is a homeomorphism of $\mathbf{R}^n$ onto itself. We may assume $p = 0 \in \operatorname{Int} D$.
Because $0$ is an interior point, an open ball $B_\varepsilon(0)$ is contained in $D$ for some $\varepsilon > 0$. The dilation $x \mapsto x/\varepsilon$ is a homeomorphism. We may thus assume $\overline{\mathbf{B}}^n = \overline{B}_1(0) \subseteq D$.
Claim. Each closed ray starting at the origin intersects $\partial D$ in exactly one point.
Let $R = \{ tx : t \ge 0 \}$ be a ray from the origin, with unit vector $|x| = 1$. The set $D$ is compact. The intersection $D \cap R$ is closed and bounded in $\mathbf{R}^n$, so it is compact. The distance function $t \mapsto |tx|$ is continuous on $D \cap R$. It attains a maximum at a point $x_0 \in D \cap R$. For $t > 1$, the point $t x_0$ does not belong to $D$. Every neighborhood of $x_0$ therefore contains points outside $D$. Because $x_0 \in D$, the point $x_0$ lies in $\partial D$.
Every point on the segment from $0$ to $x_0$ other than $x_0$ has the form $\lambda x_0$ with $0 \le \lambda < 1$. Let $z \in B_{1-\lambda}(\lambda x_0)$. Set $y = (z - \lambda x_0)/(1 - \lambda)$. Then $$ |y| = \frac{|z - \lambda x_0|}{1 - \lambda} < \frac{1 - \lambda}{1 - \lambda} = 1. $$ Thus $y \in \mathbf{B}^n \subseteq D$. The point $z$ can be written as $$ z = \lambda x_0 + (1 - \lambda) y. $$ Both $x_0$ and $y$ belong to $D$. Convexity of $D$ gives $z \in D$. This shows $B_{1-\lambda}(\lambda x_0) \subseteq D$. The point $\lambda x_0$ is an interior point of $D$. Hence $x_0$ is the only boundary point on the ray $R$.
Boundary Homeomorphism. Define $f: \partial D \to \mathbf{S}^{n-1}$ by $$ f(x) = \frac{x}{|x|}. $$ Because $0 \notin \partial D$, the function $x \mapsto x/|x|$ is continuous on $\partial D$. Each ray from the origin intersects $\partial D$ at exactly one point, so $f$ is a bijection. The boundary $\partial D$ is a closed subset of the compact space $D$, so $\partial D$ is compact. The sphere $\mathbf{S}^{n-1}$ is Hausdorff. By the Closed Map Lemma, $f$ is a homeomorphism.
Homeomorphism of the Closed Cell. Define $F: \overline{\mathbf{B}}^n \to D$ by $$ F(x) = \begin{cases} |x| \, f^{-1}\left(\frac{x}{|x|}\right), & x \neq 0, \\ 0, & x = 0. \end{cases} $$ Geometrically, $F$ maps each radial segment connecting $0$ with a point $\omega \in \mathbf{S}^{n-1}$ linearly onto the segment from $0$ to $f^{-1}(\omega) \in \partial D$.
The map $f^{-1}: \mathbf{S}^{n-1} \to \partial D$ is continuous. Thus $F$ is continuous on $\overline{\mathbf{B}}^n \setminus \{0\}$. Because $\partial D$ is compact, it is bounded in $\mathbf{R}^n$. There exists a constant $M > 0$ such that $|f^{-1}(\omega)| \le M$ for all $\omega \in \mathbf{S}^{n-1}$. For $x \neq 0$, we have $|F(x)| = |x| \, |f^{-1}(x/|x|)| \le M |x|$. As $x \to 0$, $M |x| \to 0$. This establishes continuity of $F$ at the origin.
Points on distinct rays through the origin map to distinct rays. On each ray through the origin, $F$ is a positive linear scaling. Hence $F$ is injective.
Let $y \in D$. If $y = 0$, then $F(0) = 0 = y$. If $y \neq 0$, the ray through $y$ intersects $\partial D$ at a unique point $x_0$. Then $x_0 = f^{-1}(x_0/|x_0|)$. The point $y$ lies on the segment between $0$ and $x_0$, so $y = \lambda x_0$ for some $\lambda \in [0, 1]$. Setting $x = \lambda (x_0 / |x_0|)$ yields $F(x) = \lambda x_0 = y$. Thus $F$ is surjective.
Convexity of $D$ ensures that the image of $F$ lies in $D$. The domain $\overline{\mathbf{B}}^n$ is compact and the codomain $D$ is Hausdorff. By the Closed Map Lemma, $F$ is a homeomorphism. By construction, $F(0) = 0 = p$, $F(\mathbf{B}^n) = \operatorname{Int} D$, and $F(\mathbf{S}^{n-1}) = \partial D$.
Examples
- A closed bounded interval $[a, b]$ with $a < b$ is a closed $1$-cell in $\mathbf{R}$.
- A compact convex polygon in $\mathbf{R}^2$ is a closed $2$-cell.
- A solid tetrahedron, cube, or convex polyhedron in $\mathbf{R}^3$ is a closed $3$-cell.
4. Attaching Cells along Boundaries
We formalize the construction of topological spaces by attaching cells to an existing space.
Definition (Attaching $n$-Cells)
Let $X$ be a non-empty topological space. Let $\{D_\alpha\}_{\alpha \in A}$ be an indexed collection of closed $n$-cells for some $n \ge 1$. For each $\alpha \in A$, let $\varphi_\alpha: \partial D_\alpha \to X$ be a continuous map. Let $$ \varphi: \coprod_{\alpha \in A} \partial D_\alpha \to X $$ be the map whose restriction to $\partial D_\alpha$ is $\varphi_\alpha$. The adjunction space $$ X \cup_\varphi \left(\coprod_{\alpha \in A} D_\alpha\right) $$ is said to be obtained from $X$ by attaching $n$-cells along the attaching maps $\varphi_\alpha$.
Let $Y = X \cup_\varphi (\coprod_\alpha D_\alpha)$. The subspace $X$ embeds canonically as a closed subspace of $Y$. As a set, $Y$ is the disjoint union of $X$ and the open $n$-cells $\operatorname{Int} D_\alpha$. This operation provides the inductive step for assembling spaces from cells of increasing dimensions.
Attaching Map Variations (Lee, Figure 5.2)
The map $\varphi_1$ wraps $\partial D_1$ homeomorphically onto the cylinder rim. The cell bulges outward to cap the end.
The map $\varphi_2$ embeds $\partial D_2$ along a circle on the lateral surface, forming an upward bulging dome.
The map $\varphi_3$ sends all of $\partial D_3$ to a single point $p \in X$. Collapsing the boundary gives $D_3/\partial D_3 \cong \mathbf{S}^2$, forming a spherical balloon attached at $p$.
Attaching Cells to a Cylinder (Lee, Figure 5.2)
Three closed $2$-cells $D_1, D_2, D_3$ are attached to $X$ along distinct attaching maps.
Figure 3. Attaching three $2$-cells to a cylinder with outward bulges and a pinched balloon (John M. Lee, Figure 5.2).
5. Cell Decompositions and Cell Complexes
We now formulate the dual perspective. We decompose a given topological space into open cells. A partition of a space into open cells alone does not capture the topology. For example, partitioning $\mathbf{R}^2$ into vertical lines gives a partition into open $1$-cells, but misses the planar topology. The boundary of each cell must attach to cells of strictly lower dimension.
Definition (Cell Decomposition)
Let $X$ be a non-empty topological space. A cell decomposition of $X$ is a partition $\mathcal{E}$ of $X$ into subspaces that are open cells of varying dimensions, satisfying the following condition.
For each cell $e \in \mathcal{E}$ of dimension $n \ge 1$, there exists a closed $n$-cell $D$ and a continuous map $\Phi: D \to X$, called a characteristic map for $e$, satisfying two properties.
- The restriction $\Phi|_{\operatorname{Int} D}$ is a homeomorphism from $\operatorname{Int} D$ onto $e$.
- The boundary image $\Phi(\partial D)$ is contained in the union of all cells in $\mathcal{E}$ of dimension strictly less than $n$.
For a $0$-cell $e$, the characteristic map is the unique map from a singleton space $\{*\}$ to $e$. The boundary of a $0$-cell is empty.
Definition (Cell Complex)
A cell complex is a Hausdorff space $X$ equipped with a specific cell decomposition $\mathcal{E}$ of $X$. The elements $e \in \mathcal{E}$ are called the cells of $X$.
When $\mathcal{E}$ has finitely many cells, $X$ is called a finite cell complex.
Each cell $e \in \mathcal{E}$ is an open cell, meaning that it is homeomorphic to $\mathbf{B}^n$. It is not necessarily an open subset of $X$. Because $D$ is compact and $X$ is Hausdorff, the image $\Phi(D)$ is compact, and $\Phi(D) = \overline{e}$. The closure $\overline{e}$ is not required to be a closed cell, because $\Phi$ need not be injective on $\partial D$.