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 is an open cover with a function for which: Cells are disjoint: for any distinct , . No set gets mapped to a negative number: . Cells look like balls: For any and for any there exists a continuous map that is an isomorphism and also .
A cell complex is a pair where is a topological space and is a cellular decomposition of . See also CW complex
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