By Christoph Schweigert

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The spaces D = Rn ∼ = Sn \ N are n-cells. 2. Note that an n-cell cannot be an m-cell for n = m, because Rn follows, since Rn ∼ = Rm would imply Sn−1 Rn \ {0} ∼ = Rm \ {0} Rm for n = m. This Sm−1 , ˜ n−1 (Sm−1 ) = 0. Hence the dimension of a cell is ˜ n−1 (Sn−1 ) ∼ but H = Z for all n and H well-defined. , Xi , X i ∼ X= = Rni . i∈I Here, this decomposition is meant as a set, not as a topological space. 4. 1. The boundary of a 3-dimensional cube has a cell decomposition into 8 points, 12 open edges, and 6 open faces.

Follow from the observation that for an n-cell σ we have that σ = (σ \ σ) ∪ σ is contained in X n−1 ∪ σ. 4. Induction on the dimension of the cell; then use closure finiteness and σ = Φσ (Dn ). We want to understand the topology of CW complexes. 12. 1. Cells do not have to be open in X. For example, in the CW structure on [0, 1] with two zero cells 0 and 1, the 0-cells are not open in [0, 1]. 2. If X is a CW complex and σ is an n-cell, then σ is open in the n-skeleton X n . The n-skeleton X n is closed in X.

The subspace |K| := ∪s∈K s ⊂ Rn is called the topological space underlying the complex K. Simplicial homology can be defined for a simplicial complex; it depends only on |K|. Simplicial homology has various disadvantages: 2 for example, S2 can be written as a simplicial complex with 14 simplices only (obtained from the projection of the tetrahedron to the sphere), but as a CW complex with a 0-cell and a 2-cell only. A 2-torus S1 × S1 can be written as a CW complex with 4 cells, but the smallest simplicial complex has 42 cells.