Cellular decomposition

In geometric topology, a cellular decomposition G of a manifold M is a decomposition of M as the disjoint union of cells (spaces homeomorphic to n-balls Bn).

The quotient space M/G has points that correspond to the cells of the decomposition. There is a natural map from M to M/G, which is given the quotient topology. A fundamental question is whether M is homeomorphic to M/G. Bing's dogbone space is an example with M (equal to R3) not homeomorphic to M/G.

Definition
Cellular decomposition of $$X$$ is an open cover $$\mathcal{E}$$ with a function $$\text{deg}:\mathcal{E}\to \mathbb{Z}$$ for which:
 * Cells are disjoint: for any distinct $$e,e'\in\mathcal{E}$$, $$e\cap e' = \varnothing$$.
 * No set gets mapped to a negative number: $$\text{deg}^{-1}(\{j\in\mathbb Z\mid j\leq -1\}) = \varnothing$$.
 * Cells look like balls: For any $$n\in\mathbb N_0$$ and for any $$e\in \deg^{-1}(n)$$ there exists a continuous map $$\phi:B^n\to X$$ that is an isomorphism $$\text{int}B^n\cong e$$ and also $$\phi(\partial B^n) \subseteq \cup \text{deg}^{-1}(n-1)$$.

A cell complex is a pair $$(X,\mathcal E)$$ where $$X$$ is a topological space and $$\mathcal E$$ is a cellular decomposition of $$X$$.