Cell (topology)

A cell is an open ball used as a building block in a cellular decomposition of a space. Its boundary is attached to lower-dimensional cells.

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Open ball: A space homeomorphic to the interior of a closed Euclidean ball. Each cell is homeomorphic to an open ball of a particular dimension.
CW complex: A topological space built by attaching cells in stages subject to closure-finiteness and weak-topology conditions. Cells are the building blocks in the definition of a CW complex.
Cell decomposition: A presentation of a topological space as a union of cells attached along their boundaries. A cell is one constituent of the decomposition.
Open simplex: The relative interior of a simplex, excluding all its proper faces. Open simplices are specific cell shapes used in simplicial decompositions.
Closed ball: A space homeomorphic to the unit ball, including its boundary. A characteristic map sends a closed ball onto a cell together with its attaching boundary.
Skeleton (topology): The subspace of a CW complex formed by all cells up to a specified dimension. Attaching each dimension’s cells enlarges the preceding skeleton.
Simplicial complex: A space assembled from simplices joined along shared faces according to specified incidence rules. Simplicial complexes use a more rigid kind of cell with simplex-shaped closures.
Polyhedral cell: A region in a polyhedral decomposition whose closure is a polytope. Polyhedral cells impose geometric structure beyond the open-ball condition.
Boundary sphere: The boundary of an n-dimensional ball, homeomorphic to the sphere of dimension n−1. The boundary sphere is where an attaching map connects a cell to lower-dimensional cells.
Cellular chain complex: A chain complex whose groups are generated by the cells of a CW complex and whose maps encode their boundaries. Cell boundaries induce the differentials that compute cellular homology.