In Chapter 5 of Lee's Intro to Topological Manifolds (page 130), he defines a cell decomposition as follows:
I've been struggling to properly unpack this characterization. I have the two following confusions.
a) When he says the map $\phi:D\to X$ such that $\phi|_{\mathrm{Int}D}$ is a homeomorphism between $\mathrm{Int} D$ and $e$, I'm not sure what exactly is the interior of $D$, as $D$ is simply some topological space that is homeomorphic to a closed unit ball in $\mathbb{R}^n$. Thus, in a sense, wouldn't $\mathrm{Int}D= D$ because $D$ is open in $D$ itself? But of course this doesn't make much sense.
b) When Lee says $\phi$ maps $\partial D$ into the union of all cells of $\mathcal{E}$ of dimensions strictly less than $n$, which of the following does he mean? $$\Phi(\partial D) \subseteq \bigcup_{e\in \mathcal{E}'}e\text{ such that }\mathcal{E}' = \{e\in \mathcal{E}:\text{dimension of }e\text{ is}<n\}$$ $$\Phi(\partial D) = \bigcup_{e\in \mathcal{E}''}e\text{ such that }\mathcal{E}'' \subseteq \mathcal{E}'\text{ where }\mathcal{E}'\text{ is as defined above}$$ $$\Phi(\partial D) = \bigcup_{e\in \mathcal{E}'}e\text{ where }\mathcal{E}'\text{ is as defined above}$$ Any guidance is appreciated, thanks!
(a) Note that I explained these uses of the notations $\partial D$ and $\operatorname{Int} D$ on page 129, in the paragraph just before the subsection heading. It says there that if $D$ is a closed $n$-cell, then $\partial D$ and $\operatorname{Int} D$ denote the images of $\mathbb S^{n-1}$ and ${\mathbb B}^n$, respectively, under some homeomorphism $F\colon \overline {\mathbb B}{}^n \to D$.
(b) Your first interpretation is correct (as is your third; I don't see any difference between them). The union of all cells of dimension less than $n$ is a subset of $X$, and the requirement is just that $\Phi(\partial D)$ be contained in that subset. It doesn't say anything about the image being equal to the entire subset or being equal to a union of cells.