Let me first state my definition of cell complexes.
Definition 1: Cellular decomposition of a topological space $X$ is a cover $\mathcal{E}$ of space $X$, equipped with a function deg: $\mathcal{E} \to \mathbb{Z}$, such that:
- For any different sets $e, e' \in \mathcal{E}$, we have $e \cap e' = \emptyset$
- $\mathcal{E}^{(-1)} = \emptyset$
- For each $n \in \mathbb{N}_0$ and any set $e \in \mathcal{E}_n$, there exist a continuous map $\phi: B^n \to X$, which maps the open unit ball int$B^n$ homeomorphically on set $e$ and for which $\phi(\partial B^n) \subset \cup \mathcal{E}^{(n-1)}$
Here, we use otations $\mathcal{E}_k = \text{deg}^{-1}(k) $ and $\mathcal{E}^{(k)} = \text{deg}^{-1}(\{ j \in \mathbb{Z} \; | \; j \leq k \})$.
Definition 2: Cell complex is the pair $(X, \mathcal{E})$, where $X$ is a topological space and $\mathcal{E}$ is its cellular decomposition.
Definition 3: CW complex is a Hausdorff cell complex $(X, \mathcal{E})$, such that
For each cell $e \in \mathcal{E}$, there exist cells $e_1, \ldots, e_m \in \mathcal{E}$, such that $\overline{e} \subset e_1 \cup \ldots \cup e_m$
Any subset $A \subset X$ is closed in $X$ iff for every cell $e \in \mathcal{E}$ the set $A \cap \overline{e}$ is closed in $\overline{e}$.
Now, my text provides no motivation or examples. So I was trying to think of some on my own. I would appreciate if someone explains what cellular composition of these examples looks like:
Example 1: Interval $[0,1]$. In this example, I suppose we would have $\mathcal{E} = \{ \{0\}, \{1\}, (0,1)\}$ with $\text{deg}(\{0\}) = \text{deg}(\{1\}) = 0$ and $\text{deg}((0,1)) = 1$. Conditions (1) and (2) are now clearly satisfied. Condition (3) is also satisfied; for $e = (0,1)$, the map $\phi: B^1 \to [0,1]$ can be translation plus contraction, and we have $\phi |_{\text{int} B^1}: \text{int}B^1 \to (0,1)$ homeomorphism, while also $\phi(\partial B^1) \subset \cup \mathcal{E}^{(0)}$. For $e = {0}$ and $e = {1}$ condition (3) also holds trivially. Is this reasoning okay?
Example 2: Interval $(0,1)$ (and similarly $[0,1)$). I don't know how one could find a cellular decomposition for this space, because of condition (3); I think the boundary of $B^1$ would not map in $\cup \mathcal{E}^{(0)}$. So, what to do here? In general, how to take care of open spaces?
Example 3: A cube with a dome on top, or any slightly more complicated object. How to find a cellular decomposition of such a space?